Not an electronics person in any way. I've heard of "AC Ripple" with CB Power Supplies. You could set your DVOM on an AC setting, put it across a 12 Volt DC circuit, and see traces of AC Voltage. That may be the possibility mentioned above.
Remember, it's my belief that If GOD wanted me working on Electrons He would have made them Big Enough to See.
In the simpler electric (vs electronic) world I function a little better in, I have a nagging worry that the GROUND circuit has a high resistance (loose or corroded) somewhere. Yes, the Exterior 12 VDC Marker Lights, Turn/Stop/Tail get the HOT side from the tow vehicle and House Circuit (like Inverter) from the House Battery. BUT they share the ground side.