Democracy: A system of government in which every citizen is entitled to participate equally in government.

Republic: A system of government by the citizens, as opposed to a monarch, emperor, or dictator.

Constitutional republic: A republic in which the powers and functionining of government are restricted by the constitution.

I don't necessarily think "democracy" and "republic" are mutually exclusive.

I do think "democracy" and "constitutional republic" are exclusive.

For example:

In a democracy, someone else's wish to confiscate my property counts equally with my wish to keep my property.

In a constitutional republic, the government's powers are limited by the constitution, providing protection for citizens against arbitrary governmental powers.

Of course, you could have a constitutional republic with a poorly-functioning constitution, which in effect functions more and more as a democracy. So there are certainly mixed cases.

Some people would resist the definitions I have given, because I define concepts so clearly. But that is the very point of concepts: to capture the essence, not to capture the mixed cases. And objective concepts are a necessary part of rational thought.