Turn signals cause all rear lights to flash?

This sounds like a bad ground at the rear of the truck, or the trailer wiring connector is shorted out. Here are the wiring diagrams so you can see how the system works and the ground location so you can check it out and fix the problem.

Ground location - on spare tire hoist bracket, upper left kick panel

Check out the diagrams (below). Please let us know what you find. We are interested to see what it is.