With a POD no earlier than 1775, how could the Ottoman capital city, Istanbul (called Kostantiniyye at the time) become permanent territory of the United States by 1950?
The territory doesn't have to be intended to be permanent when initially acquired, but perhaps a situation like WW2 and the Cold War, with a large ideologically opposed bloc, requires it to remain under direct administration.
Bonus points if Constantinople is eventually admitted into the Union as a state.