The Genoese were historically involved with trade between the Eastern Roman Empire and the west. At various points, they would receive rights to establish settlements and were even endowed with fiefs by the emperors. The Genoese nobleman Manuele Zaccaria received Phocea as a hereditary fief from Emperor Andronikos II Palaiologos in 1275, with Phocea remaining in his family's possession until it was repossessed by the Byzantines under Andronikos the third Palaiologos in 1340. Phocea is adjacent to Chios( the island you mentioned from EU IV) in the Anatolian coast. Unbeknownst to everyone at that point, their fates would be interlinked for more that 2 centuries. In Genoa proper a company was chartered as a compensation for services to the city by various citizens and nobles which stipulated that its shareholders would be supported in the attempt to recapture Phocea and conquer nearby Chios and would thus have exclusive rights to mercantile and various other business activities in the area. That company was named "Maona di Chio e di Focea" and would come to rule the region from 1346 to 1566. The company's shares were eventually bought by the Giustiniani family, which granted them definite control of the island and the city. They only lost that when the Turks conquered the island in 1566, because it had become a refuge for fugitive ottoman slaves and a safe harbor for Christian fleets at arms reach from the ottoman capital, Constantinople. As far as finances are concerned this company made profit real soon and was a major reason for the prosperity in the area; trade flourished and the produce, such as wine, cloth, salt, olives, alum, spice and the native Mastiha, a resin which is extracted from trees native only to the island of Chios found their way to lucrative new markets. This rejuvenated the islands economy and along with a retention of prior privileges and a respect of the local customs and religion( eastern orthodoxy) won the Giustiniani family and the shareholders prior to them the loyalty and cooperation of the local elit and to some extent the rural Greek population. There are records of mixed marriage between Genoese settlers and Greek natives and at the peak of their influence, Genoese amounted to approximately 20% of the islands population. At times the natives would become disillusioned with their foreign elit, which along with some discrimination on accounts of religion, heightened tensions, but it never reached a breaking point. Moreover, the city benefited from the company's activities because of taxation that would eventually find its way to Genoas coffers. There was even some sort of government appointed bureaucracy at place in the area to ensure compliance by the company and that due procedure is followed. Last but not least, a member of the Giustiniani family, Giovanni, has become a legend in Greek society because he participated, of his own volition, to the defence of Constantiple in 1453, personally outfitting and training 700 Greeks and Genoese from Chios and leading them in the Empire's final days. He was such a Charismatic leader that eventually he rose to a vital commander for the Greek defense and held off repeated ottoman assaults until his ultimate demise in the city's defense.

So, I want to develop some of the points made by u/LeNoirClement while adding a certain degree of chronological structure to develop a better understanding of the way Genoese institutionalism evolved in the eastern Mediterranean.

So, I really think the crucial moment in the story of Italian colonization in the eastern Mediterranean in the second half of the Middle Ages is the war of San Sabas (1256-1261 with raiding afterward) is the crucial moment. From 1299 onward, Genoa, Venice and Pisa existed in a state of semi-peaceful commercial rivalry over control of commercial enclaves in the Crusader States. They all had colonies in the major ports of the Levant. But due to random circumstances, what might be understood as an inevitable war broke out in Acre in 1256, and the Genoese were essentially driven out of that southern zone of the eastern Mediterranean. I think it's this moment that you start to see the Genoese look to the Aegean and this other commercial zone.

When I say "Commercial Zone" (tm), I am thinking in terms of the triple route to Europe from east and south Asia.Though antiquated to a certain extent, the breakdown of commercial routes in the global Middle AGes in Before European Hegemony is solid. You have goods transferred through the Red Sea and Egypt, across the Middle East and through the Levantine ports, and in the north through central Asia and through the Black Sea and Constantinople. And after the establishment of Venetian hegemony in the Levantine route and, to a a certain extent, a strong hand in then Alexandrian/Egyptian trade—at least until the Mamluk period—the Genoese were left seeking fortunes in the northern route.

In the late 13th century you have a much more disorganized approach with what I would describe as semi-noble freebooters trying to get theirs in the eastern Mediterranean. Benedetto Zaccaria got in good with the Roman emperor and kinda grifted control of Phocaea. The island was primarily important for its alum mines, but the island was crucial on the island hopping route out of the Black Sea and into the MEditerranean basin.

This period of expansion happened during an era when the traditional nobility and semi-nobility were in control of the Genoese state. The establishment of the colony at Caffa, which was actually established with the backing of the Genoese government and was structured differently. IT was essentially established with the support of the state, and I think it was guided by the wealthy merchants who were not explicitly noble or semi-noble. This assumption is based on the names included in the government lists at Caffa. Control of the Black Sea area was completely consolidated in the aftermath of the 4th crusade. As venice almost dominated the zone after aiding the crusaders, the Genoese allied themselves with the exiled Paleologoi kingdom at Nicea and aided in their reconquest of Constantinople. From there they became the dominant commercial power of the aegean and the Black sea, at least until 1296. From there, the Genoese and Venetians fought over the stopover points from the black sea (chios, etc) and the rest of the Mediterranean. This was a down and dirty narrative, and I'd be happy to explain more if necessary.