Our births were registered with the US consulates. My birth certificate is 4 pages long. But because we were born in foreign countries we could claim citizenship of those countries. At the time, the US didn’t recognize dual citizenship. This process was to prevent any confusion.
If I recall correctly, it was not necessary to have a passport to enter countries where we had a SOFA. You just showed your military ID when entering and leaving. And US passport control also accepted the military ID.