rear light goes out when lights are on

Exactly.

If the Running Lights are on, and the brakes are applied, the Running Lights turn off on ONE side.

He SAID the brake light goes off, but technically the running light is the one that goes out. What really happens is that the light that is already on goes off.

Yup, like I said at the outset...classic symptoms of a bad ground. No matter how the tail lamp assembly is grounded...right now, it isn't.

-- Old Crow '82 FLTC-P "Pearl" '95 Wrangler YJ TOMKAT, SENS, BS#133, MAMBM, SLOB#13, DH#2

You have one of two problems. The ground connection to the base of the bulb itself is s***ty, or the ground connection from the base of bulb to the frame/boddy is s***ty.

There are two sources of ground to investigate. Does the bulb itself ground through the base? And, does the base ground to the frame/body?

One test is to take the working bulb from the other side and move it to the failure side, and vice versa, to see if the problem follows the bulbs. Often, the bulb itself is the problem, even when you _think_ you put in a good one.