In such cases, there is usually a “consular report of a birth abroad” that the embassy sends to Washington and the parents get a copy of. Military servicemembers also frequently get this. It’s not required to be done by any means and it is not a birth certificate though it can usually be used in place of one.

But! I think it may be the only document issued about a birth that is proof of US citizenship. And, paradoxically, is not available to people born inside the US.