I think that the connection to the rear fog light might work the opposite from the way you assume.

Since the rear light only works when the head lights are on, this circuit recieves a signal or some voltage to indicate that the headlights are on - perhaps the rear light even gets its power from the headlight circuit.

I think that the rear light connection to the fog lights must operate in the same manner as the connection to the headlights, to allow the rear light to operate if the fog lights are on.
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2 8s & 2 7s 600,000 miles total