That is the whole point of discussion .. You can have a Islamic Theocracy but should not name it as "Islamic Republic" ... Because the use of word "Republic" may result in confusion for others as well as for our own people ...
For example :
The verb "to table" has different meanings in American & British English (opposite)
in AmE it means "to suspend from consideration"
In BrE it means to "raise for consideration"
During World War 2 a meeting of top Allied commanders nearly broke up in a furious row because both Brits and Americans wanted urgent action on something but one side's expressing this as tabling it was misunderstood by the other side as meaning doing nothing!
So what we mean when we say "Republic"(Islamic) is not how the rest of the world understands the meaning of word "Republic" .... We should avoid using it ..
If I am not mistaken, Islam does not specify the structure of the state or how the people select their leader. It is something open for the people to figure out themselves, which is often developed organically by societies. In this regard Islam is very flexible, by staying silent.
this "theocracy" is not something , which Islam tells you that you "must" have as a condition for being a good Muslim or what not. But a society is free to develop ANY form of governmental structure or state.
This can either be theocracy, republic, democracy, communism, aristocracy, socialism,tribal society, nomadic society, even the concept of "state" is something optional, and not necessary.
It is simply up to the people/group to decide what sort of a social organization they need to develop to regulate the behaviour of their society to have order.
Islam is very flexible and silent on that.
How the prefix Islam come into play is by becoming a guideline for these "systems". It can do so by placing certain limitation and/or by encouraging certain narratives.
E.g It is Islam to let minority non Muslims to practice their religion and give them the freedom to go to their worship places of choice.
Thus it becomes a narrative these "systems" would follow, either it be a socialist state, a republic, a communist state, a aristocratic state, a theocracy, kingship, a tribe, or a nomadic group.
Thus having a republic is not against Islam. Neither these two are oxymoron.
But rather Islam compliments republic quite beautifully.
In case of relation between Islam and republic. Leader explained it quite nicely.
Republic is actually a nation-state federation or union binded by the constitution. such as Pakistan or America.
the word Islamic republic is A muslim nation state which is binded by a constitution governed by the principle of Islam.
In a republic, people can select their representative and these representatives would select the leader. This leader can be anyone, he can be either Hitler, who represents the will of his people to kill all jews, or he can be Nelson Mandela, who wants inter ethnic peace.
However, by complimenting this republic with Islamic guidelines, you can prevent a hitler from killing all the jews, and encouraging Nelson Mandela in his quest for inter ethnic peace.
So the idea of Islamic Republic does exist, and is even better than a normal republic.