My backup lights only work when I have one bulb or the other installed but not both. The other 4 mini lights in the same circuit on the trunk work. The backup lights do not work at all when the headlights are on. Could it be the backup light relay and where is that located? I checked wiring on the trunk and multiple bulbs.
My guess would be that you have a bad ground someplace in the trunk. Is there a place where there is a connection to the body of the car in the trunk? If so, remove the connection and scrape off any corrosion,
I agree with Triedaq. From what you describe it does appear to be bad grounding causing the trouble. The more lights you have on the circuit, the more the voltage drop appears across the resistance of the bad connection to ground due to the added current flow in the circuit. That is why a second bulb makes them go out. Most of the supply voltage gets dropped across the bad ground connection instead of the lights.
I also agree with the above. Bad grounds can cause some unusual problems
I found it. It was a broken ground wire where the trunk meets the body in the shielded wires.
Good deal. Thanks for the update.