I have two furnaces and two a/c units. Zone 2 is the bedroom and has a furnace and a/c unit. With the thermostat on Auto, they fight each other. For example, on a cold night we set the thermostat on Auto at 70F. The furnace comes on and the single furnace vent blows very hard and HOT!. The remote thermocouple is located on the bedroom wall right under the furnace vent. So it heats up very quickly and when it hits 72F, the a/c comes on to cool it off. It then cools it quickly to 68F and the furnace comes on again. When this happens, air is coming out of the a/c and furnace vents simultaneously for a little while as the furnace blower continues to run for a bit after the flame is out and similar on the a/c unit's fan.
For now, we just do not run in Auto. Thought about a vent deflector, but due to the location of the vent, it won't really help. So just no Auto mode for the bedroom. Auto in the main living area works semi okay, aside from running the heat pump until the exterior temp is below 30F (which is crazy, it loses its efficiency below about 40F). We tend not to run that zone in Auto either because of that.