Can you be a democratic republic? they are two different styles of government after all. so where is this politically confused place meant to be?
They are not mutually exclusive. Republic simply infers that a country's governmental authority is derived from the people as opposed to a hereditary ruler, Monarch or divine leader.
Both the US and China are Republics yet have very different systems of Governance because the method by which power is derived from the people differs greatly.
The word Democratic can be used to broadly explain the form of political structure in place, where the leader and/or leading party are directly elected by the people. For a real world example see the Democratic Republic of Congo.
However, having Democratic in a country's name does not necessarily mean it is a democracy, for instance the official name of North Korea is actually the Democratic People's Republic of Korea, even though it is actually a one party state.