Someone born on a military base overseas (outside of U.S. territory), is issued a Consular Report of Birth Abroad form FS-240 if the parent(s) registered the birth at a local consular office or embassy. If the parent did not register the overseas birth, then in order to obtain proof of citizenship, one would need to file a Certificate of Citizenship (form N-560) which is issued by the Immigration and Naturalization Services.
It's necessary for those procedures because the 14th Amendment doesn't apply to those military bases not on soverign U.S. territory therefore require laws created by Congress to allow the parent(s) citizenship to pass to their child.
Some of those bases are (at least partly) controlled by the laws of the host country, the details of which are worked out in the SOFA agreement(s).
As an example, Royal Air Force Station Upwood in England, is owned by the UK's Ministry of Defense, yet is operated as a U.S. military base.
If a child is born on that base in England, the parents of that child must register for a British birth certificate (in addition to registering a Consular Report of Birth Abroad to obtain U.S. citizenship):
In case you didn’t know, all newborns need to have an American and British birth certificate. You need to register your infant in the British system within 42 days of birth. Your baby will need to be registered in the district of birth.http://www.501csw.usafe.af.mil/shared/media/document/AFD-120514-018.pdf
Another example:
United States Naval Hospital Yokosuka, Japan
Information for Birth RegistrationImmediately after birth, parents fill out the “Information for Birth Registration” form and submit it to Birth Registration, Patient Administration, located on the 3rd floor.
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The hospital birth certificate is the document you will need to apply for your child’s official certificate, which is called the “Consular Report of Birth Abroad” (FS-240). The hospital birth certificate is an unofficial document but may be used to update the sponsor’s page 2 and also for DEEPS enrollment. The FS-240 will be the official record of your child’s birth as an U.S. citizen. While it is not a birth certificate, the FS-240 does serve many of the same functions as a birth certificate. You will also need the FS-240 when applying for your child’s social security card.