interestingly a Roman Catholic Patriarch of Jerusalem would pose the lovely theological issue of, presumably the greeks asking, "why exactly did you cut off contact with us if your still gonna go for the Pentarchal structure".
Technically Islamic Republics in almost all cases aren't theocratic, much like how Israel isn't run by Rabbi's even though it's a Jewish State.
Iran, well together with the Vatican and the Tibetan Government in Exile, are very much the only actual theocracies.
Even then Iran is Elective, to a point.
The Vatican is an Elective Monarchy, and Tibetan GiE is a Reincarnatic Theocratic Monarchy.