I got a detailed report, but it only said things such as "checked plug ABC, pin 123 for continuity”. They found continuity in a pin, which presumably indicated a short circuit. They removed the plug, cleaned it and refitted it - no more short.
Athe whole problem was that the rear washer pipe has a joint in the most stupid location, pouring water onto the Body Control Module and immobiliser. I'm not sure if either of these would cause the car not to turn off, but possibly the BCM?
Despite fitting cable ties to hold the pipe to the joint, mine came apart again in last week's cold weather. That was despite having more or less neat screen wash fluid, so I'm sure that the real issue is that the rear screen was nozzle is getting wet and icing up. The washer motor tries to pump fluid, the pressure in the pipe builds up and the joint splits.
Come the spring, I'm going to see if I can fit a continuous length of washer pipe from the 'engine bay' to the tailgate. Then, if the pipe pops a joint, it'll be well away from any vital electronics!