The terms “Islamic” and “Republic” do not go together.
It would be more accurate to refer to the Islamic “Republic” as a theocracy, or by the term so often used in the past, a caliphate. The current term is probably a perversion of the original definition of “republic” as a form of government, because that suggests there is some degree of legitimacy to the current regime in Iran.
The Iranian regime is no more legitimate than the North Korean regime, and for a similar reason. Both are imposed by force, unto threat of death for dissidents, and sustained by daily reinforcement of that threat.