CALABRESI, Circuit Judge:
Plaintiffs-Appellants Chad E. Cohen and Kirsten Cohen (the “Cohens”) filed a Complaint in the Superior Court for the State of Connecticut against Defendant Third-Party-Plaintiff-Appellee Postal Holdings, LLC (“Postal Holdings”), stating a claim of private nuisance based on injuries allegedly caused by Postal Holdings’s failure to maintain a property it owned that was adjacent to the Cohens’ home. Postal Holdings then filed a Third Party Complaint against the United States Postal Service (“USPS”), to which it had leased the relevant property, stating claims of common-law indemnification and contractual indemnification.
The USPS removed the suit to the United States District Court for the District of Connecticut (Thompson, /.), pursuant to 28 U.S.C. § 1442(a)(1), and then moved to dismiss the Third Party Complaint against it for lack of subject-matter jurisdiction. On January 15, 2015, the district court granted the motion, but, without expressly addressing the question, retained supplemental jurisdiction over the Cohens’ remaining state-law claims against Postal Holdings. By Order of June 1, 2016, the district court granted summary judgment in favor of Postal Holdings on the Cohens’ state-law claims. The Cohens now appeal that Order.
Because a federal district court cannot exercise supplemental jurisdiction over state-law claims unless it has subject-matter jurisdiction over the federal claims originally presented, we VACATE the district court’s June 1, 2016 Order and REMAND with instructions to remand the state-law claims to the state court for further proceedings consistent with this opinion.
I. BACKGROUND
A. Factual Background
The Cohens own a house located at 30-32 Catoonah Street in Ridgefield, Connecticut. Next to the Cohens’ property, at 26-28 Catoonah Street, are two lots owned by Postal Holdings.
The USPS is the lessee of 26-28 Catoo-nah Street under a ground lease entered into by Postal Holdings’s predecessors-in-interest and the USPS in 1982. Pursuant to a 2006 amendment to the 1982 ground lease (collectively, the “Lease”), the USPS has possession and control of 26 and 28 Catoonah Street. The Lease provides that the USPS, “at its own cost and expense, shall construct and maintain all buildings, structures^] and improvements on the ... premises” and that the USPS’s “responsibility for maintenance shall be fulfilled at such time and in such manner as [the USPS] considers necessary.” App’x at 38-39 ¶ 9. The Lease also provides that the USPS will indemnify Postal Holdings “from all claims, loss, damage, actions, causes of action, expense[,] and liability resulting from the use of the ... property.” App’x at 38 ¶ 8.
The USPS owns and operates a postal facility on the 26 Catoonah Street property. The 28 Catoonah Street property, which abuts directly upon the Cohens’ property, contains a physical structure that was, since at least 2006, in a state of disrepair. Kirsten Cohen testified that she told the manager of Postal Holdings, Lisa Quattrocchi, in early 2008 that “there were beer bottles strewn everywhere, shingles flying off in the wind, broken glass, and ... the shrubbery just being overgrown, the grass not being taken care of.” App’x at 137. Kirsten Cohen also testified that, in response, Quattrocchi said “she would never want her children growing up next to a home like that” and that Kirsten Cohen was “left with the impression that [Quat-trocchi] was taking care of it,” notwithstanding that Quattrocchi had explained to her that 28 Catoonah Street “was being leased by” USPS at the time. App’x at 138-39. Over the course of an exchange between the Cohens and Quattrocchi that spanned several years, Quattrocchi sent multiple letters to the USPS expressing her concerns about the condition of 28 Catoonah Street. The state of the property, however, apparently did not improve.
B. Procedural History
On October 21, 2013, the Cohens filed a Complaint against Postal Holdings in the Connecticut Superior Court, stating a claim of private nuisance based on Postal Holdings’s alleged failure to abate the conditions at 28 Catoonah Street, and seeking at least $15,000 in damages, exclusive of interest and costs.
On May 29, 2014, Postal Holdings filed a Third-Party Complaint in the Connecticut Superior Court against the USPS, im-pleading it into the action. The Third Party Complaint stated two claims against the USPS—common-law indemnification (Count One), and contractual indemnification (Count Two)—without conceding that there was a private nuisance.
On June 4, 2014, the USPS removed the action to the United States District Court for the District of Connecticut, pursuant to 28 U.S.C. § 1442(a)(1).3
On June 20, 2014, the USPS moved to dismiss the Third Party Complaint against it for lack of subject-matter jurisdiction, pursuant to Federal Rule of Civil Procedure 12(b)(1). The USPS argued that the claims against it fell within the purview of the Contract Disputes Act, 41 U.S.C. §§ 7101-7109 (“CDA”), which grants the United States Court of Federal Claims exclusive jurisdiction over contract disputes with executive agencies of the federal government, such as the USPS. . .
On July 5, 2014, the Cohens filed a First Amended Complaint.against Postal Holdings in the district court, restating their claim of private nuisance, adding a separate claim for negligence, .and restating their claim for damages. The 1982 ground lease and 2006 amendment áre appended to the Cohens’ First Amended Complaint.
On January 15, 2015, before any discovery had been undertaken, the district court granted the USPS’s motion and dismissed the Third Party Complaint pursuant to Rule 12(b)(1). Cohen v. Postal Holdings, LLC, No. 14CV800, 2015 WL 225051, at *2-*3 (D. Conn. Jan. 15, 2015).
Having dismissed the claims against the USPS for lack of jurisdiction, the district court nonetheless exercised supplemental jurisdiction oyer the Cohens’ remaining state-law claims and, on June 1, 2016, granted Postal Holdings’s motion for summary judgment on those claims. The record does mot1 indicate that either of the parties objected to the district court’s decision to exercise supplemental jurisdiction over the remaining state-law claims.4 Nor do either of the parties' now appeal that decision, or the district court’s antecedent dismissal of the Third Party Complaint for lack of subject-matter jurisdiction. The Cohens, instead, appeal only the district court’s Order granting Postal Holdings’s motion for summary judgment on the state-law claims.
II. DISCUSSION
“[EJvery federal appellate court has a special obligation to satisfy itself not only of its own jurisdiction, but also that of the lower courts in a cause under review .,.. ” Bender v. Williamsport Area Sch. Dist., 475 U.S. 534, 541, 106 S.Ct. 1326, 89 L.Ed.2d 501 (1986) (internal quotation marks omitted). “And if the record discloses that the lower' court was without jurisdiction,” the appellate -court has “jurisdiction on appeal, not of the merits but. merely for the purpose of correcting the error of the lower court in entertaining the suit.” Id.
Thus, before considering the Co-hens’ appeal on its merits, we must first determine whether the district court properly asserted supplemental jurisdiction over the Cohens’ state-law claims against Postal Holdings. To the extent that this inquiry involves a question of subject-matter jurisdiction, we determine nostra sponte whether the district court properly exercised its authority. Joseph v. Leavitt, 465 F.3d 87, 89 (2d Cir. 2006). “The standard of review for determinations regarding subject-matter jurisdiction is clear error for factual findings, and de novo for the legal conclusion as to whether subject matter jurisdiction exists.” Lyndonville Sav. Bank & Tr. Co. v. Lussier, 211 F.3d 697, 701 (2d Cir. 2000).
For the reasons explained below, we hold that the district court correctly determined that the CDA deprived it of subject-matter jurisdiction over the claims in the Third Party Complaint. The district court therefore correctly dismissed the Third Party Complaint pursuant to Rule 12(b)(1), Havingdismissed all federal claims for lack of subject-matter jurisdiction, however, the district court erred in retaining supplemental jurisdiction over the Cohens’ remaining state-law claims,
A. Dismissal of All Federal Claims Pursuant to Rule 12(b)(1) Bars Supplemental Jurisdiction over Related State-Law Claims
Under 28 U.S.C. § 1367(a), “in any civil action of which the district courts have original jurisdiction, the district courts shall have supplemental jurisdiction over all other claims that are so related to claims in the action within such original jurisdiction that they form part of the same case or controversy ..,. ” Id. (emphases added). This means that a district court “cannot exercise supplemental jurisdiction unless there is first a proper basis for original federal jurisdiction.” Nowak v. Ironworkers Local 6 Pension Fund, 81 F.3d 1182, 1187 (2d Cir. 1996). It follows that when a district court correctly dismisses all federal claims for lack of subject-matter jurisdiction pursuant to Rule 12(b)(1), the district court is thereby precluded from exercising supplemental jurisdiction over related state-law claims. See 28 U.S.C. § 1447(c) (“If at any time before final judgment it appears that the district court lacks subject matter jurisdiction, the case shall be remanded.”); Nowak, 81 F.3d at 1187.5
In assessing the correctness of the district court’s exercise of supplemental jurisdiction over the Cohens’ state-law claims, wé therefore must first determine whether the district court correctly dismissed the Third Party Complaint pursuant to Rule 12(b)(1) for lack of subject-matter jurisdiction, or whether it should instead have dismissed the Third Party Complaint pursuant to Rule 12(b)(6) for failure to state a claim. ■
“In most circumstances, it makes little practical difference whether the district court. ... labels its dismissal of an action as one for lack of subject matter jurisdiction under Rule 12(b)(1) or for failure to state a claim under Rule 12(b)(6).” Id. at 1188. The distinction, however, matters here. When a district court dismisses, pursuant to Rule 12(b)(6), all claims over which it properly has original subject-matter jurisdiction, the district court may still, under § 1367(c), exercise its discretion to retain supplemental jurisdiction over related state-law claims. Valencia ex rel. Francos Lee, 316 F.3d 299, 306 (2d Cir. 2003).
In Nowak, we noted that a district court has no such discretion when it properly dismisses, pursuant to Rule 12(b)(1), all federal claims over which it- might have had original subject-matter jurisdiction. Upon reviewing the .district, court’s dismissal of the plaintiffs federal claims in that case, however, we found that the district court had incorrectly determined that it lacked original, subject-matter jurisdiction over the federal claims, and that it therefore had improperly dismissed those claims under Rule 12(b)(1). Nowak, 81 F.3d at 1190. We also found that dismissal of the federal claims would nonetheless have been appropriate under Rule 12(b)(6), and that it would not have been an abuse of discretion for the district court to have-then retained supplemental jurisdiction over the related state-law claims. Id. at 1191-92. We hence reconstrued the district court’s dismissal as a dismissal pursuant to Rule 12(b)(6), and affirmed the district court’s exercise of supplemental jurisdiction. Id. at 1193; see Jims v. Pension Plan of Make-Up Artist & Hairstylists Local 798, 170 F.3d 162, 165 (2d Cir. 1999) (similarly reconstruing a 12(b)(1) dismissal as a dismissal under 12(b)(6)).
As a result, it is important to determine whether the district court in this case, like the district court in Nowak, dismissed all federal claims pursuant to Rule 12(b)(1) based on an incorrect determination that it lacked subject-matter jurisdiction over, the federal claims. If so, then the district court’s exercise of supplemental jurisdiction over the Cohens’, state-law claims might still have been proper under § 1367(c).
We therefore proceed, below, to assess the correctness of the district court’s determination that it lacked subject-matter jurisdiction over the Third Party Complaint. Doing so requires us to address several issues, some of which raise legal questions that have not yet been answered by this circuit. We address each of these issues in turn.
B. The District Court’s Jurisdiction over the Third Party Complaint under the Postal Service Reorganization Act and the Contract Disputes Act
“[T]he Postal Service is part of the [Federal] Government and that status indicates [sovereign] immunity unless there is a waiver.” U.S. Postal Serv. v. Flamingo Indus. (USA) Ltd., 540 U.S. 736, 744, 124 S.Ct. 1321, 158 L.Ed.2d 19 (2004). Thus, after the action was removed to the district court pursuant to 28 U.S.C. § 1442(a)(1), the district court was obligated to determine whether Postal Holdings’s claims fell within a specific waiver of the USPS’s sovereign immunity. See Makarova v. United States, 201 F.3d 110, 113 (2d Cir. 2000) (“The doctrine of sovereign immunity is jurisdictional in nature, ■ and therefore to prevail, the plaintiff bears the burden of establishing that her claims fall within an applicable waiver.”) (internal citation omitted).
Postal Holdings argued before the district court that the Postal Reorganization Act of 1970, 39 U.S.C. § 101 et seq. (“PRA”), provides such a waiver. In the PRA, Congress “waive[d] the immunity of the Postal Service from suit by giving it the power ‘to sue and be sued in its official name.’ ” Flamingo Indus., 540 U.S. at 741, 124 S.Ct. 1321 (quoting 39 U.S.C. § 401(1)). Moreover, the PRA provides that “the United States district courts shall have original but not exclusive jurisdiction over all actions brought by or against the Postal Service.” 39 U.S.C. § 409(a). Postal Holdings therefore contended that the PRA not only waived the USPS’s sovereign -immunity, but also gave the district court jurisdiction over its Third Party Complaint. •
As the'USPS pointed out, however, the PRA’s jurisdictional provision stands in tension with the CDA, which gives exclusive jurisdiction to the Court of Federal Claims over contract disputes with the federal government in excess of $10,000. See. 28 U.S.C. § 1346(a)(2) (“[T]he district courts shall not have jurisdiction of any civil action or claim against the United States founded upon any express or implied contract with the United States or for liquidated or unliquidated damages in cases not sounding in tort which are” subject to review under the CDA); Up State Fed. Credit Union v. Walker, 198 F.3d 372, 374-75 (2d Cir. 1999) (per curiam).
The CDA creates a remedial scheme whereby persons wishing to bring a contract-based claim against the federal government must first submit their claim in writing to a contracting officer, 41 U.S.C. § 7103(a)(1)-(2), who “shall issue a decision in writing,” id. § 7103(d). A “contracting officer’s decision on a claim is final and conclusive and is not subject to review by any forum, tribunal, or Federal Government agency,” except as authorized by the CDA itself. Id. § 7103(g). The CDA provides that, within 90 days of receipt of a contracting officer’s decision, an aggrieved contractor may either appeal a decision by a contracting officer to the relevant board of contract appeals, see id. § 7104(a), or “bring an action directly on the claim in the United States Court of Federal Claims,” id. § 7104(b)(1). The CDA “established” a board of contract appeals within the USPS called the Postal Service Board of Contract Appeals. 41 U.S.C. § 7105(d)(1). In each instance, further review may be had in the United States Court of Appeals for the Federal Circuit. See id. § 7107(a)(1)(A).
Although courts have characterized the CDA’s remedial scheme as “the paradigm of a ‘precisely drawn, detailed statute’ that preempts more general jurisdictional provisions,” A & S Council Oil Co., Inc. v. Lader, 56 F.3d 234, 241 (D.C. Cir. 1995) (quoting Brown v. GSA, 425 U.S. 820, 834, 96 S.Ct. 1961, 48 L.Ed.2d 402 (1976)), our court has not yet determined whether the CDA overrides the PRA’s grant of jurisdiction to the federal district courts. See 39 U.S.C. § 409(a).
In Anselma Crossing, L.P. v. United States Postal Serv., 637 F.3d 238 (3d Cir. 2011), the Third Circuit addressed this question, and determined that the CDA is a statutory bar to the jurisdiction granted to federal district courts by the PRA. It gave several reasons for its holding. First, the CDA was enacted moré recently than the PRA. Id. at 246. Second, the terms of the CDA are “more precisely drawn than the terns of the PRA.” Id.; see Cmty. Health Care Ass’n of N.Y. v. Shah, 770 F.3d 129, 157 (2d Cir. 2014) (“It is a basic principle of statutory construction that a specific statute controls over a general provision.”). Third, the CDA does not exclude USPS contracts even though it specifically excludes several other types of contracts from its coverage. Anselma Crossing, 637 F.3d at 246. Fourth, permitting contract claims against the USPS to proceed in federal district court would undermine the CDA’s purpose of collecting contract claims against the government in a consistent and expert forum. Id.
A majority of other federal courts of appeals that have addressed this question have reached the same conclusion as the Third Circuit. See Goodin v. U.S. Postal Inspection Serv., 444 F.3d 998, 1001-02 (8th Cir. 2006); United States v. J & E Salvage Co., 55 F.3d 985, 988-89 (4th Cir. 1995) (finding contract actions against USPS barred from district court without mentioning PRA); Jackson v. U.S. Postal Serv., 799 F.2d 1018, 1022 (5th Cir. 1986) (stating that the enactment of the CDA in 1978 removed' district courts’ concurrent jurisdiction over suits in contract against the USPS); cf. Campanella v. Commerce Exch. Bank, 137 F.3d 885, 890-92 (6th Cir. 1998) (holding that sue-and-be-sued clause in Small Business Act (“SBA”) did not grant district court jurisdiction over contract claim); A & S Council Oil Co., 56 F.3d at 241-42 (same). But see In re Liberty Constr., 9 F.3d 800, 801-02 (9th Cir. 1993) (holding SBA’s sue-and-be-sued clause provided district courts an independent statutory grant of jurisdiction); Wright v. U.S. Postal Serv., 29 F.3d 1426, 1429-30 (9th Cir. 1994) (applying reasoning of Liberty Construction to the USPS based on 39 U.S.C. § 401(1)).
We find the reasoning of the Third Circuit in Anselma Crossing persuasive, and join the position of the majority of circuit courts on this question. We therefore hold that the district court correctly determined that the CDA grants the Court of Federal. Claims exclusive jurisdiction over contract claims.against the USPS, notwithstanding the PRA’s general grant of jurisdiction in 39 U.S.C. § 409(a).
C. The Applicability of the Contract Disputes Act to the Third Party Complaint
We next consider whether the claims stated in the Third Party Complaint fall within the purview of the CDA, and thus beyond the jurisdiction of the district court. In. order to answer this question, we must ask whether the particular contract involved in this matter (ie., the Lease) is covered by the CDA.
. The CDA governs “express or implied contract[s] ... made by an executive agency for ... the procurement of property, other than real property in being.” 41 U.S.C. § 7102(a)(1) (emphasis added). Although our court has not yet determined whether leases of real property fall within the scope of this exclusion, we find persuasive the Federal Circuit’s determination in Forman v. United States, 767 F.2d 875, 879 (Fed. Cir. 1985) that .they do not.
' In Forman, the Federal Circuit interpreted the legislative history of the CDA, as well as its text, and concluded that the “real property in being” exclusion “refers to the procurement of existing,interests— fees, easements, leases, etc.—and not the initial creation of these interests.” Id. The Federal Circuit thus interpreted the CDA to draw a distinction between the government’s acquisition of pre-existing property interests—through, for example, the power of eminent domain—and the government’s creation of new property interests through contract. The court reasoned that “by entering into a lease' the Government does not acquire a pre-existing interest in the land [but rather] establishes a new one,” and therefore that the CDA applies to leaseholds created by contracts with executive agencies. Id.; see Jackson, 799 F.2d at 1022 (citing Forman for the principle that a contract for lease of office space, in which the USPS was lessee, was covered by the CDA); Salt River Pima-Maricopa Indian Cmty. v. United States, 86 Fed.Cl. 607, 613-16 (2009) (using the distinction Forman made between acquiring an existing property interest, and entering into a new one, when interpreting 41 U.S.C. § 7102(a)); 1-10 Indus. Assocs., Inc. v. U.S. Postal Serv., 133 F.Supp.2d 194, 195 (E.D.N.Y. 2001) (“The [CDA] applies to leases of real property.” (citing Forman)); cf. United States v. Triple A Mach. Shop, Inc., 857 F.2d 579, 585 n.2 (9th Cir. 1988) (finding that a lease of land in which the Navy was lessor did not fall within the CDA’s provision governing the procurement of property, other than- real property in being, 41 U.S.C. § 7102(a)(1), “because here the government is not procuring property; it already owns the property”).
In: the case before us, the original ground lease signed in 1982. by the USPS and Postal Holdings’s predecessors-in-interest indicates that the USPS’s leasehold interest in 26 and 28 Catoonah Street was created through that contract. Therefore, applying the Federal Circuit’s reasoning in Forman to the facts of this case, we find that the Lease is governed by the CDA,
We must still determine, however, whether Postal Holdings’s claims in its Third Party Complaint truly arise out of the Lease or whether-—as Postal Holdings' argued before the district court—its claim for common-law indemnification actually sounds in tort rather than contract. If Postal Holdings is correct, then the CDA’s jurisdictional bar would not apply to its claim. See 28 U.S.C. § 1346(a)(2) (depriving the district courts of jurisdiction over claims against the government “founded upon any ,.. contract with the United States .in cases not sounding in tort” (emphasis added)). ‘ ,
In Up State Federal Credit Union, we held that the question of whether an action is.“‘at its essence’ a contract action .... depends both on the source of the rights upon which the plaintiff bases its claims, and upon the type of relief sought.” 198 F.3d at 376 (quoting Megapulse, Inc. v. Lewis, 672 F.2d 969, 968 (D.C. Cir. 1982)). Where • the right the plaintiff seeks to vindicate “stems from no independent, non-contractual source” and the remedy for violating that right is essentially a contractual remedy arising out of a contract within the scope of the CDA, then the claim is subject to the CDA notwithstanding a plaintiffs attempt to disguise it as a tort. Id. at 375-77.
Applying the analysis in Up State Federal Credit Union, we find that the claims' in Postal Holdings’s Third Party Complaint are both essentially contract claims arising out of the Lease.
With regard to the claim for common-law indemnification—Count One of the Third Party Complaint—Postal Holdings alleges the following:
8. Pursuant to the ... .lease, USPS, as lessee, had exclusive control over the property located at 28 Catoonah Street, which is where the plaintiffs allege a dangerous and defective condition occurred.
9, Although the third party plaintiff, Postal Holdings, does not concede a private nuisance, if anyone was negligent, it was USPS for failing to prevent and/or abate an alleged nuisance at 28 Catoo-nah Street.
App’x at 15 (emphasis added). As a remedy, Postal Holdings requests “[a]n order that, pursuant to the terms of the lease, the .... USPS is required to indemnify Postal Holdings for any award of .damages.” App’x at 16 (emphasis, added). Therefore, the right upon which Postal Holdings bases its claim for common-law indemnity, as well as the remedy it requested, arises from contract.
The same is, obviously true of Postal Holdings’s claim, in Count Two, for “contractual indemnification.” App’x at 15 (emphasis added).
We conclude that the district court correctly determined that the claims in the Third Party Complaint against the USPS fall within the scope of the CDA, and therefore are not within the district court’s subject-matter jurisdiction.
D. The District Court’s Dismissal of the Third Party Complaint Pursuant to Rule 12(b)(1)
It is well-established that when the CDA bars parties from bringing claims in a federal district court, such claims are properly dismissed for lack of subject-matter jurisdiction, pursuant to Rule 12(b)(1). See Up State Fed. Credit Union, 198 F.3d at 377 (affirming a district court’s dismissal of a claim for lack of subject-matter jurisdiction because “the CDA ... provides the sole basis for waiver of sovereign immunity in this case-and therefore ... the Court of Federal Claims has exclusive jurisdiction over this matter.”); see also Anselma Crossing, 637 F.3d at 240; Goodin, 444 F.3d at 1000-02; B & B Trucking, Inc. v. U.S. Postal Serv., 406 F.3d 766, 768 (6th Cir. 2006) (en banc); J & E Salvage Co., 55 F.3d at 989-90; cf. O’Rourke v. Smithsonian Inst. Press, 399 F.3d 113, 122 (2d Cir. 2005) (affirming a district court’s dismissal of an action against the Smithsonian for lack of subject-matter jurisdiction on the grounds that the Court of Federal Claims had exclusive jurisdiction). Hence, given the applicability of the CDA to the Third Party Complaint, we find that the district court correctly dismissed the Third Party Complaint pursuant to Rule 12(b)(1). Because Postal Holdings had not taken the required threshold step under the CDA of •presenting its claims to the contracting officer, 41 U.S.C. §§ 7103(a), 7104(b), transfer of the Third Party Complaint to the Court of Federal Claims is not appropriate.
III. CONCLUSION
Under the CDA, the district court lacked subject-matter jurisdiction over Postal Holdings’s Third Party Complaint. The district court therefore correctly dismissed the Third Party Complaint pursuant to Rule 12(b)(1). Having properly dismissed Postal Holdings’s Third Party Complaint, the district court lacked supplemental jurisdiction over the Cohens’ state-law claims.
There being no jurisdiction in the district court over the Cohens’ suit, we VACATE the district court’s Order granting summary-judgment to Postal Holdings on the state-law claims, and REMAND with instructions to remand the state-law claims to the state court for further proceedings consistent with this opinion.