By rear running light do you mean tail light? Have you done some electrical work lately? If so the best course of action is to check the last thing you did. Do you have access to an electrical schematic for the bike and also a digital multimeter? Those are two must have items for electrical troubleshooting.
no electrical work whatsoever... when i got the bike it was "rough". Actually thats to nice of a word, it was a pile... did not run, so there was no testing anything...
Took apart and restored...
The harness was intact and complete and there was a plug with a corresponding end for everything, so no guess work for wiring this thing and that is why this is so odd...
The more I read the more I think it may be a relay as I took off the ignition switch and with the key in on position I switched the 56 and 58 wires.... the headlight turned off and the running light came on along with the gauge light...
This is a used/replacement ignition switch as the bike didn't come with keys and i couldn't find a locksmith who could cut a key for the ignition as none of them wanted to take it apart...
so for now daytime riding is good... night time, not so much.