A Westerner permanently living in the Eastern Mediterranean is not a very common phenomenon nowadays, but in the past sizable communities of Westerners existed in cities like Istanbul (Constantinople), Izmir (Smyrna), Beirout, Cairo and Alexandria. These Levantines1, as they were called, were usually descendants of traders from various backgrounds and in some cases even of the inhabitants of the Crusader states.
Pay attention to the fact that they weren't called Orientals2, which was a term to denote the native population. However, they weren't considered Westerners either and were actually explicitly distinguished from Westerners temporarily residing in the Levant. While remaining attached to the culture they brought from their country of origin, many of the Levantines had also adapted to their societies, fluently speaking the language, being familiar with the local customs and values and even marrying locals, mostly Christians like Greeks, Armenians or Maronites. Due to their foreign background and often also because of their occupations, the Levantines had strong ties with the West (and in particular with their country of origin). Yes, the world of nations was not fully developed yet, but they were already fairly transnational.

A Levantine family in Istanbul
source: www.levantine.plus.com
A similar migration occurred in the second part of the 20th century, yet in the opposite direction. Immigrants from the Levant initially arrived as 'guest-workers' in Western European countries, but they soon decided to settle here with their families. They too established sizable communities and have become an essential part of Western European societies. They are fluent in the languages, have adopted many of the customs and values of their host countries and have also married locals. They have maintained their ties with their country of origin and have travelled back and forth on a regular basis. In the course of time they have even adopted the nationality of their host countries, while still keeping their original one. In this respect, these immigrants have been, just like the Levantines, both bicultural and transnational. In fact, they could be considered the Levantines of today! 3
Even though the two immigrant communities differ from each other with regard to social class, they have very much in common as mentioned above. The Western immigrants were no 'Westerners' anymore, but neither were they 'Orientals'; they were Levantines. Similarly, the Eastern (and Southern) Mediterranean immigrants are no 'Orientals' anymore, but neither are they 'Westerners'; they must be Ponentines4 then!
1. From Italian 'Levante' and French 'Levant' meaning 'East', or more precisely: 'The lands where the sun rises'
2. From Italian 'Oriente' and French 'Orient' also meaning 'East' or 'The lands where the sun rises'.
3. This is obviously also true for Eastern Mediterranean immigrant communities in the Americas.
4. From Italian 'Ponente' and French 'Ponent' (archaic?) meaning 'West' or 'The lands where the sun sets', after the example of 'Levantine'.