He was born on a base which is US Soil though, was he not?
Unfortunately, no! The hospital (only hospital available for his mother to give birth) was not on US soil. Obama, Hillary and a bunch of other democrats passed some sort of bill/amendment or whatever you call it to bypass that particular issue - sorry I don’t have the link. I think they thought that if BO ended up having an issue that the Republicans would kindly do the same thing. But this bypass is unconstitutional.
From the foreign service manual.
7 FAM 1116.1-4 Not Included in the Meaning of "In the United States" (TL:CON-64; 11-30-95) a. A U.S.-registered or documented ship on the high seas or in the exclusive economic zone is not considered to be part of the United States. A child born on such a vessel does not acquire U.S. citizenship by reason of the place of birth (Lam Mow v. Nagle, 24 F.2d 316 (9th Cir., 1928)).
b. A U.S.-registered aircraft outside U.S. airspace is not considered to be part of U.S. territory. A child born on such an aircraft outside U.S. airspace does not acquire U.S. citizenship by reason of the place of birth.
c. Despite widespread popular belief, U.S. military installations abroad and U.S. diplomatic or consular facilities are not part of the United States within the meaning of the 14th Amendment. A child born on the premises of such a facility is not subject to the jurisdiction of the United States and does not acquire U.S. citizenship by reason of birth.