Name a communist Islamic state, now or in the past.
There were and are plenty of the "stans" which were and are under Soviet/Russian rule with very large Muslim populations.
The point is, Islam and communism function in a very similar matter, a central governing body, with a single all powerful leader/dictator (Imam/Caliphate) which controls all aspects of a society.
There are plenty of Islamic countries which have a democratic government of sorts which shouldn't exist either by your definition.