During the Cold War, most likely the 1950s and beyond. I’ll let the rest up to you since you’re a much greater expert in this topic than me obviously.
Okay so Let's say that the Cyprus Crisis escalates into full-scale war. Turkey and Greece are now fighting a war in 1974. Turkey has a leg up on Greece. Larger population, and closer to Cyprus. Turkey instantly mobilizes all local troops to Cyprus to force the Greeks off the Island. Soon both countries are moving troops. Greek Islands near Mainland Anatolia are taken through week-long seiges. Greece starts making more defences to the Greek islands. Turkey sends the majority of their troops to take Cyprus. The Turks quickly Overrun Cyprus and is under full military control.
Now Comes Foreign Intervention. During the Crisis in our timeline, the U.S put an embargo on Turkish arms sales. So now that its' escalated, America embargos all imports, and exports from Turkey. America openly supports Greece in this conflict. The Soviet Union seizing on this event openly supports and blockades Greece imports and Exports.
Now that Turkey seized Cyprus, Greece and America support a Kurdish revolution in Kurdish Turkish areas. Kurds from all over the Middle East, From Iran to Armenia come to fight for a free Kurdistan. And to make American-Turkish relations even worse, America recognizes the Armenian genocide.
So now we got Kurdistan vs Turkey vs Greece. How can we make it worse? Well after America recognizes the Armenian genocide, The Soviets start sending troops to fight Kurdish rebels, and restore the land to Turkey, while Turkey deals with the Greeks.
Now how can we make this worse? Well, we could get Mehmed Ertuğrul Efendi son of Mehmed VI last crowned Sultan of the Ottoman Empire, seeing a chance to gain prestige goes to Istanbul, and forms a rebellion in the city to reform the religious Ottoman empire.
Soooooo there
We got
Greece vs Turkey vs Neo-Ottoman empire in Istanbul vs Kurdish rebels vs the Soviet Union vs America.
Now go crazy.