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This is something I'm thinking of incorporating into my ongoing TL:
In 1871, the US had a little rout with the Joseon in trying to establish diplomatic relations that ended in failure, despite having a pretty one sided victory.
What if, following the "battle", the US decided to occupy the island? They may have, initially, only the smallest demand that the island be returned only when diplomatic relations are established with which the Koreans can ask for it.