Stuart, here's what's probably happening: The lack of a ground between the bed and the frame is preventing either one of the filaments from grounding through the brass bulb base as they should. When the voltage cannot find its normal ground, it backs up through the other filament, and finds its ground by running backwards through what should be a hot wire. When the lights are turned on, both wires have voltage, preventing any path to ground. It's a very common problem, one that few mechanics, even ones with extensive electrical experience, really understand. The simple way to troubleshoot it is to run a long wire from a known good ground and touch it to the brass base of the bulb. If the lights work normally, you've found the problem. The permanent fix is to run a heavy ground wire from the bed to the frame, and make sure the taillight assemblies are well-grounded to the bed.
Jerry