I assume most would retake Anatolia 400 years of cultural diffusion had already occurred between Greeks and Turks
the folk costume of Corfu never conquered by Turks.
" A head-cloth ("phakiolion" or "maphorion") which ranged from a simple cloth coming from below the helmet (as still worn by Orthodox clergy) to something more like a
turban, was standard military headgear in the Middle and Late Empire for both common troops and for ceremonial wear by some ranks " -
https://en.wikipedia.org/wiki/Byzantine_dress "The origins of turbans are uncertain. It is typically agreed by historians that it emerged from the Middle-East. The Islamic prophet, Muhammad, who lived 570–632, is believed to have worn a turban in white, the most holy colour. Many Muslim men choose to wear white turbans as well as green turbans, because it represents paradise, especially among followers of
Sufism. Turbans became prominent in
Al-Andalus when Moorish men wore them. It is also worn across North-Africa. In parts of North Africa, where blue is common, the shade of a turban can signify the tribe of the wearer.
[5] A style of headware called a
phakeolis which was worn by Greeks since the 10th century
[6][7] was adapted to a more turban style under
Ottoman rule, in which a piece of cloth around it was placed. It was still worn by their Greek-speaking
descendants in the early 20th century."
Greece
"In
Greece, specifically the island of
Crete, the men traditionally wear a turban known as a
sariki. The headwrap's name is borrowed from
sarık, the
Turkish word for turban. Today, it may be more commonly known as a
kritiko mandili (Cretan kerchief). It is not found commonly amongst the younger generation, but mostly worn by older men in remoter, mountainous villages. It is traditionally black, the color of mourning, but can also be found in white, for festive occasions such as weddings. It is very commonly found as a part of the standard Cretan costume worn by members of Greek dance groups. A similar head garment is also worn in various parts of Cyprus, and can be seen in various Cypriot shows depicting traditional Cypriot society, e.g. the comedy Aigia Fuxia. And in ancient times
Greeks, wore a distinctive style of turbans.
[8]"
-https://en.wikipedia.org/wiki/Turban
Ottoman Turk Osman I wearing a turban. The turban was worn by both Byzantine men and women, and when the Byzantine Empire was conquered by the Ottoman Turks, the Turks too began wearing the turban.
"A headdress with ancient roots, the turban is made from a long strip of cloth, most often cotton or silk, which is wrapped around the head, usually in a specific pattern. The turban frequently covers the whole head, concealing the hair from view, and sometimes the cloth is wrapped around a turban cap rather than directly around the head. Some experts believe that the turban originated in Persia, modern-day Iran, while others think that it was invented by the Egyptians. However, the use of the turban first became widespread during the years of the Byzantine Empire (476-1453 C.E.), and since that time turbans have been strongly identified with Eastern cultures and religions.
The Byzantine Empire was characterized by a blend of Eastern and Western cultures, and one symbol of this blending was the adoption of the Persian turban by Emperor Constantine (c. 285-337 C.E.). The turban was worn by both Byzantine men and women, and in 1453, when the Byzantine Empire was conquered by the Ottoman Turks, the Turks, too, began to wear the turban. Though turbans often have great religious or political meaning in the cultures in which they are worn, during various periods certain Westernized turbans have become popular as women's fashion accessories."
http://www.martelnyc.com/middle-ages/headwear-of-the-byzantine-empire.html
FOR MORE INFORMATION
Houston, Mary G. Ancient Greek, Roman, and Byzantine Costume and Decoration. Lanham, MD: Barnes and Noble, 1977.
Tulips, Arabesques, and Turbans: Decorative Arts from the Ottoman Empire. New York: Abbeville Press, 1982.
as for the fez:
"The fez was introduced into the
Balkans[
citation needed], initially during the
Byzantine reign, and subsequently during the Ottoman period where various
Slavs, mostly
Bosniaks and
Serbs, started wearing the head-wear " -
https://en.wikipedia.org/wiki/Fez If anything the Turks look Greek.