So, is she a "natural born" citizen? Would John Jay consider her to be a foreigner?
If she was born at the Navy Base and got a US Consulate Infant Born Abroad Birth Certificate and Passport then yes, of course she is. If She was born in a Japanese Hospital on the local economy and was not reported to the State Department at the time of her birth, then no, she is not. The difference is that by Status of Forces treaty our overseas posts and bases are considered US territory effectively
7 FAM 1133.3-3 What Constitutes U.S. Physical Presence d. Interpretation of "Periods of Honorable Service in the Armed Forces of the United States" (4) Only periods of honorable U.S. military service abroad count as periods of physical presence in the United States. However, some persons who have received other than honorable discharges may have some periods of honorable service that can be confirmed by the military authorities. Other members of uniformed services are considered U.S. Government employees pursuant to 5 U.S.C. 2105(a).
g. Interpretation of "Dependent Unmarried Son or Daughter and a Member of the Household"
(1) General Interpretation. A citizen son or daughter of any parent whose
employment abroad with the U.S. Armed Forces, the U.S. Government, or
a designated international organization qualifies as physical presence in the
United States may count as physical presence in the United States any
time spent abroad with such parent during the parent's employment as
long as the son or daughter was an unmarried, dependent member of the
parent's household. Whether the parent was a citizen or an alien at the time
of employment is immaterial.
http://www.state.gov/documents/organization/86757.pdfEarlier in this thread I mentioned that my Daughter was Born in the Azores. The State Department made it clear that I needed to register her birth as a Georgia Citizen. When we got back home, I took the paperwork to the Floyd County, GA Registrars Office and had her put in the books as a resident of Georgia!
We were told at the time that the Portuguese law indicated that she could elect to have Portuguese citizenship when she came of age but of course she never did that.