The LEDs should only light with current in one "direction." If the potential in the frame rises above the level of what it sees through the relay, then there shouldn't be any light given off. At least that's the theory.

I am wondering where the wires heading off to the right go though. As you say, maybe that's the dash indicator. Adding a power diode in that line, before the signals, might be enough to stop it.

Right - since diodes are basically electrical one-way valves & LED's are Light Emitting Diodes.

The branches headed to the right are to the dash indicator(s), I expect - so if there's a back-feed voltage induced in the opposite circuit, it would be in the right 'direction'. iirc, typical gate voltage for led's is about 3.5, so it wouldn't take much.

I am still going with poorly installed or poor quality additional components of what was installed during these mods. If there is a flasher relay, lose the relay and see if it lights up the other side when going super fast blinking. If it doesn't, get some resistors and call it a day.

That's where I think that adding another power diode, in the "correct direction", would do the trick. It should drop the voltage across the indicator LEDs to below threshold. It's either that, or start monkeying around with a Zener circuit to act as a switch.

Another view for you Papa. This one is a bit different since it is the Aussie version. I didn't notice but it is simpler. The Euro version had wires (the ones missing from this diagram) going to an alarm system that must have been standard there. Apparently the service manual I have is for Euro and Aussie bikes, not USA. The labels for the new stuff are below.

25 Combination Meter, whatever that is:shrug:
26 Hi Beam Indicator Light
27 Turn Signal Indicator Light
28 Meter Lights
32 Fuel Sender
49 Oil Level Switch

A quick search shows that "combination meter" means your dash display (speedo, tach). It sure looks like leakage through the diode pairs, in the dash display. They've got some bizarro bridge rectifier deal going on there. There wouldn't be enough current to light up an incandescent lamp, but diodes are a different story. Like I said adding a power diode, or a whopping big power resistor, would likely do the job.

Bingo (I think) - those little triangles with the bar in front are symbols for diodes, and with a circle around it is an LED. So it does have a single indicator light, right? The four diodes arranged at the indicator lamp allow the signal current to pass through the indicator & to the bulbs through one pair or the other depending on which side is switched on - you can trace it by the direction of the arrows. The 'opposite' pair of LED's acts to block the flow of current to the un-switched signal side - but they aren't completely blocking & allowing enough voltage through to dimly light the opp. side.
Increasing the inline resistors @ the signals should take care of it, I think.

Sounds right. The term is "Leakage Current."

Nah, not bizzaro - pretty typical diode application, similar to how the AC current from the stator is 'rectified' to DC current by only allowing the 'positive' voltage through (but with only 2 legs instead of 3).

I say 'bizarro' because, if you follow the path that current would take (well actually opposite to the path, since flow is from negative to positive opposite to the old theory, that all of the component symbols were designed to follow), current feeds from the display straight back into the opposite signal through that bridge. Dumb, if you're designing for an LED system, but they weren't designing for LEDs.