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"Turkish" as in "Republic of Turkey", 'cause Ottomans ain't no Turkish nationals
With a PoD no earlier than 1935, make Dodecanese by annexed into Republic of Turkey by 1955 at the latest.
However, if you'd choose a post WWII PoD, then Turkey can get it in anytime before 1990. The sooner the better, though, and personally with this kind of PoD I'd like to see Turkey has it already by 1965 but I'll leave it to the takers.