In 1970 my daughter was born in Schweinfurt, West (at the time) Germany to myself, an American citizen, and her mother, a German citizen (at the time). I immediately went to the American consulate in Munich and had her, my daughter, "made" an American citizen, complete with US passport. One trip to the consulate, and a few bucks, is all it took. Two months later we all came to the US and lived happily (mostly) ever after. No immigration file.
You don’t think they made a copy of her passport and supporting documentation?