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The tail/brake lights on my '77 don't work when the headlights are on, but work with the headlights off. All the other lights work though.
This is bad at night -- the rear end of my car is dark when I'm stopping.
Check the ground wires at the taillights, I believe there is one on each side. Check the contacts in the bulb sockets, and check the large block connecter on the drivers side of the firewall that goes into the fuseblock (mine was covered in a gooey black tar-like substance). :U
It sounds like you have a poor connection either to the taillights or at the ground for the taillights, without the headlights on your system has enough amperage to overcome the resistance, but with the headlights on the available amperage is not quite enough.