You should check the integrity of the wiring to the backup lamps.
Easiest way is to construct a test lead long enough to easily go (no stretching or tight bending) to the transmission switch or fuse block. Ideally you would put an alligator clip on one end, you can leave the other end bare, and just use your DMM when at the rear / open trunk of the vehicle. But if you secure the bare wire with electrical tape instead of an alligator clip, that should be fine, it's just that less secure connections when they are remote can be problematic so insure you have a good remote connection.
Any wire long enough you happen to have around will be fine for the voltage and continuity tests you will make in automotive troubleshooting. If you don't have any, buy some cheap trailer hookup wire (so-called "Primary Wire") at an auto parts store or hardware store. Any reasonable size (gauge) will do for the brief testing you will perform. Suggest 18GA.
You should see the expected result (12V across the terminals for each lamp, or none across the terminals) with the two modes (reverse selected + some other gear / neutral selected).
If there are two lamps, check across them (ie check the positive terminal at the L side lamp and the negative terminal at the R side lamp with both lamps installed in the sockets). In some cases a lamp failure on one side will kill both lamps.
Removing the lamp housings (they should attach with removable connectors of some kind) and giving them a good cleaning can often fix these kinds of mysterious auto lamp issues. It can be the problem even when your continuity tests say it shouldn't be. Just not worth it to be chasing down some phantom issue when you can start with the cleaning and know for sure.
Hunting down and fixing electrical gremlins only happens two ways. Either you follow a comprehensive checklist of steps or you are experienced at it and your intuition and observation can often quickly find an issue. But even an experienced troubleshooter, if his instincts don't pan out quickly, will go back to the step-bu-step procedure. So that's what I recommend you do.