Try this..
I know this is an older post, but I just read it in the newest TDR magazine.
Had this problem, and the solution for me was new tail light socket boards (both sides). Very easy to replace, and quite inexpensive if you shop around.
I paid under $30 for both including shipping to Alaska at amazon.
I tend to get lots of condensation in my tail light assemblies due to the environment here, and suspect they became corroded inside the boards. The connectors were also very corroded, but even after a thorough cleaning, the reverse lights still would not light.
Replaced with new boards, and presto. All is well.