AHC: Korean Hokkaido

Preferably with a PoD later than 1000 CE, how can Korea colonize Hokkaido in the same way Japan did in the 1800's?
 
Japan essentially conquered Hokkaido before the 1800s, the 19th century was just the nail in the coffin. Anyways IMO this is very difficult with this late a POD. I'd say that a Korean Inner Asian empire is a prerequisite.
 
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