I'm not quite sure when the Turks invaded, but ...
AIUI pretty much ALL the people who now live there are either immigrants from Turkey, or were born, after the invasion, to immigrants.
The region has been pretty much ethnically cleansed of true Cypriots, Greek and Turkish alike. The place is run FOR immigrants BY immigrants. And that is why most Cypriots (INCLUDING Turkish Cypriots) would object.
The same is true of large chunks of the West Bank. And if you go back in history you will find that, sadly, this is common. Even in my own neck of the woods you find that history has been rewritten to hide this fact - the original IRA was largely *Protestant*, it was Northern Ireland that was the hotbed of revolt against the English Crown, etc etc. You'll see similar incongruities in Scotland and Wales - the majority of "Welshmen" are cornish, most "Scots" are Picts, etc etc.