I agree that a monist can have a map. But you said "consciousness exists on the map" which (AFAICT) would imply the map to be "real" else how would something happen on it as opposed to being an emergent property of the contents of the territory?

For example temperature is an emergent property of matter, it does not happen on the map, do you agree? Assuming you do, what would it mean for temperature to happen "on the map"? (Obviously that's nonsensical so for a second here just don't think about what temperature actually is.) Would such a state of affairs not imply dualism?