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I'm currently writing and improving a TL where Korea becomes a Republic very early on.
In order to balance it budget for modernisation it attempts to make some concessions, including Jeju Island.
It is this right here:
The AHC is as follows: make the randomest, most unexpected country to buy this island by 1899. That excludes China, Japan, Britain, Russia, France, and the US.
One example I was actually thinking of was the Philippines buying it, etc.
POD is 1875.
Go crazy.