I don’t know about that. I am an American citizen and my wife is Japanese, and my daughter was born in Japan. She received CITIZENSHIP (albeit dual Japanese/American) and has an American overseas birth certificate.
However, if Obama forged this, he should be ushered out of the race for committing a felony.
Were you in Japan as US military or is there a consular agreement that covers this?
They had no trouble being recognized under the law as American citizens.
There are international treaties that regularize this stuff, as well as status of forces agreements, and US laws governing the situation. Those laws may well vary by the part of the world involved.