"Become orthodox or die"

The phrase is translated "Orthodoxy or Death", and the meaning is not "Be Orthodox or Die", rather "I would either rather be Orthodox or Dead". It is just a rendition of a common Orthodox Christian phrase out of various Saint's Lives stories out of the Turkish Rule, where martyrs who were being tortured for the sake of being converted, would just say that "They were born Christian, they want to die Christian".

In cases where they were persecuted by Papal Christians during the Latin / Frankish Occupation, this would be more clearly stated as "They were born Orthodox, they want to die Orthodox". Often it was also described as "They were born Roman / Greek, they want to die Roman / Greek", though this usually had a religious connotation, as in "Greek Orthodoxy" and the "Rum Millet" as opposed to the Roman Greek nation. Of course, when uttered by an Orthodox Christian Roman Greek it could also have an ethnic dimension as well; for example, in one of the most famous use of that term, Athanasios Massavetas Diakos, who died in April 1821 AD, whose famous last words were "Roman born am I, Roman shall I die", which is later often rendered as "Greek born am I, Greek shall I die". Though there were also lighter versions of that, such as in the Orthodox-Islamic debate between imperial translator Panagiotis Nikousios "Mamonas" and the Vani Mehmed Efendi, the Islamic teacher of Ottoman Sultan Mehmed IV, in the late 17th century AD, where under pressure of the latter to become Muslim, the first responded, "I am born Christian, I want to be Christian".

I am actually against such displays of Greeks in the Hagia Sophia exacly for this reason, that this allows for people to heavily misunderstand this phrase, or even deliberately misrepresent it by twisting it as some sort of motto displaying genocidal intent of the Greeks against the Turks, or of the Orthodox against the non-Orthodox, where it is essentially exactly the opposite statement.