Brake lights do not work when headlights are on

This is actually a real common symptom caused by a broken ground circuit. When the tail lights are turned off, and you apply the brakes, current goes through the brake light bulb, can't find its path to ground, but finds an alternate path doubling back through the tail lights, over to the front running lights, through them and then to ground at the front. When the tail lights are on and you apply the brakes, 12 volts is seen on both sides of both bulbs. The difference in voltage is 0 volts, so the bulbs go out.

The ground is listed as "B7" and is shared with the license lamps. This is not going to be a problem with the bulbs' sockets because that would affect just one of them, not all of them. Instead, we have to look for what they have in common, and that isn't very much. The ground wires for all of the rear bulbs are spliced together, then a single wire travels up to the center console and is bolted to the body sheet metal next to a computer module. Unless someone was in there and unbolted those wires, that is not a likely place to find the cause of the break. You'll be more likely to find the break at the rear. If someone added a trailer wiring harness years ago and used Scotch-Lock connectors, suspect a corroded wire. Those connectors do not seal out moisture. Also look for badly worn carpeting in the rear indicating people have been stepping on wires that run through there.

Thank you Cardiodoc for your detailed response. I will search for those corroded wires and see if that does it.

Please keep us posted on your progress.