The traditional form of church-state tension in Islam is there is some monarch who is supposedly religiously legitimate, but some in the provinces do not recognize that legitimacy - because of a doctrinal dispute, or allegedly having become corrupt or worldly, or charged with innovation and heresy, etc. Some self appointed holy man or sect then revolts. If successful they set up a new monarchy, either in place of the old or more often as a split away separate state.
Successful generals occasionally try to piece the bits back together again through worldly power, or uniting against an outside threat or opportunity. They get the real power and sometimes leave a figurehead with religious legitimacy - like Shoguns and Emperors in Japan, or "mayors of the palace" and weak kings in medieval Europe, or prime ministers and ceremonial heads of state. Sometimes such "sultans" manage to re-establish religious legitimacy for themselves and eclipse the old figurehead. Sometimes they bring only part of the old state along, and others object to a perceived lack of religious legitimacy in it all, all over again.
Everyone deferring to the Caliph without demur because he is the head of the religion as well as of the state, has not actually happened in practice since the first century of Islamic history. Omar (the second Caliph) was assassinated by a foreign slave. His successor was killed by a Muslim mob in Medina. The fourth, Ali, was assassinated by a Kharijite, a sect that rejected all three rival rulers of their day and wanted no overlord but God. They claimed the real caliphate had ended with Omar. Shiites did not recognize the next.
If this is everyone deferring to the same head of state as supposedly also the head of the religion, what would endless faction over sectarian differences look like? ;-)