Capitalism is primarily an economic system, not a political philosophy. And while it requires property rights in order to function, it is primarily concerned with solving problems in the absence of coercion, so it is absolutely compatible with anarchy.
You’re making a fundamental error when you think that property rights would not or do not exist in anarchy. What doesn’t exist in anarchy is the enforcement of such rights by a STATE. A property owner (or in this case, really anyone who lays claim to a property, since a state that could issue official deeds does not exist) still has the right to defend their property using violent means if necessary.
So yes, capitalism and anarchy are absolutely compatible.