PeterT

The best way of thinking about what is happening is to think of the way you would arrange for a relay to latch itself ON and not turn off.  The way this is achieved is to connect up the contacts so that if you energise the coil, it switches connects a live (12V) feed to the side of the coil being energised.  So, when the initial coil energisation is removed, the relay stays energised directly from the battery.  This is what is happening on the V11.  The lights are connected directly to the battery; they don't go through the ignition switch but, if everything works correctly, the ignition switch energises the lights relay and this switches on the lights and off when the ignition is switched off.  What is happening is that a short circuit between this circuit and the output of the ignition switch is keeping the lights ON AND energising the output of the ignition switch and keeping the engine running.  In effect it is just like the simple latching relay circuit.  And the only way to switch it off is to interrupt the coil current which pulling in the clutch and pressing the starter button does.  In fact, if I'm right, you don't need the tank and to start and run the engine to get this effect.  Just switch on the ignition.  The lights will come on but if you switch off the ignition, the lights will stay on.  Assuming this happens, I think you can be confident it is a short between the red/black wire and the Orange/blue wire that is causing the problem.
Now, your comment that if you disconnect the 5-way everything works OK is also interesting.  The red/black wire goes through pin 4, but the Orange/Blue wire that goes to contact 3 of the ignition switch does not.  However, if if you have the side-stand down, the Orange/Blue wire could be falsely energised if there was a short-circuit between the red/Black and the white wire that goes between the Neutral gear relay and the 5-way connector and this also connects to the White/brown of pin 2 of the 5-way if the run-stop switch is closed.    This would imply that the fault between these two wires is not in the long harness down the spine of the bike to the relays in the rear but in the wiring and switches of the RH handlebar controls, either in the 5-way connector, the wiring or the control switches themselves.  This is good news.  Can you put a meter across the various pins in the 5-way connector when it is disconnected?  I'd specifically look to see what pin 4 (Red/black) seems to be connected to.  It should connect to pin 5 if you collapse the side-stand, but should not connect to any of the other 3 wires. My expectation is that you will find a spurious connection between the red/black on pin 4 and the adjacent White/ Brown of pin 2 or possibly the white wire of pin 1 that are connected together by putting the kill switch into the RUN position.  If so you then need to work out if the problem is a fault in the RH handlebar switches, in the connector or the wiring between them.