Had the same exact problem, it wound up being my trailer hitch plug getting caked with mud.
When headlights are off, the taillights work perfectly, with lights on, all three come on.
If you have a factory, or some sort of hitch, disconnect the plug from the main loom and see if it helps. If it does, then clean out the plug, if not, look elsewhere.
My Jeep is a chick magnet with reverse polarity.
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2000 WJ, 4.5l, AX15, 242 D60/14b, 5.5" lift front, 7.5" rear. Ford Expedition rear coils, Ram 350 shocks, JKS, IRO Long Arms, cool Badlands flag thingy.
1987 XJ - Gone but not forgotten