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Good trouble shooting, maybe a voltage drop problem, look for a bad ground for those lamps, or maybe a B+ feed problem, like corroded pin/socket connection, or splice to the rear lamps, such that both lamps load cause the battery voltage to drop across the faulty connection.
If you have a multimeter, perform some voltage drop tests, to determine if you have some bad connections. If you've not done voltage drop testing, there are instructions in the "Battery Tutorial" sticky atop this forums index page.
If you don't have a multimeter, you could rig a ground test jumper lead, one end to a good clean unpainted body spot, the other end as a test probe to provide a parallel ground connection to each lamp & maybe able to narrow down the faulty ground question, as your probe would cause both to light up if you have a faulty wiring ground.
You could do the same with a fused jumper for the B+ feed, if you come to suspect it.
Let us know how your trouble shoot goes.
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