It was possessed by many different entities over the centuries, but we are interested in how Venice held it before the Battle of Chioggia, part of the conflict between Venice and Genoa. The Treaty of Turin in 1381 that ended the war handed Tenedos over to Count Amadeus VI of Savoy, who negotiated the treaty. He then destroyed any fortifications on it so that it could not be used for military purposes. He also evacuated all the Venetians and others, who were to be compensated financially. Genoa was supposed to pay for the effort to destroy the structures.
It was only a decade after the Treaty that concerns arose about the spread of the Ottomans. The island was completely deserted and without any functional buildings, so there were discussions about re-occupying and re-fortifying it.
Genoa and Venice were still the two greatest maritime republics with lots of ships going in and out of Marmara and the Black Sea. Genoa invoked the Treaty of Turin and put its foot down against any occupation by Venice. Venice likewise said they would not agree to any suggestion of shared management. Venice also spoke against granting rights to the papacy, the Knights of St. John, or the Byzantine Empire. (A case of "If I can't play the game my way, I'll prevent anyone else from playing.")
The disputes stretched out until it was too late and the Ottoman Turks had spread to the whole area. Genoa was able to maintain the biggest share of Black Sea trade up to and even after the Fall of Constantinople in 1453.
I mentioned yesterday that Amadeus of savoy was chosen to mediate between Genoa and Venice because of his ability to deal with conflicts, and that some of those were within his own family. Tomorrow I'll tell you about Amadeus and his numerous conflicts.

