You nailed it. I found your solution after doing some searching (after, of course, I posted the question....) and it was the relay.Have a look at your starter relay (Not the solenoid). Either the contacts are stuck together or following a jetwash or lots of rain water has got into it. Stat by unplugging the relay and turning your ignition switch ON, is it still doing it?, if not proceed to examine the relay.
The starter relay is plugged-in right next to the flasher relay under the left cover. Unplug it, lever off the cover with a small screwdriver or penknife and operate the armature while observing the contacts. If full of water dry it out before re-fitting.
If you prefer to replace it there are replacements available cheaply from car accessory shops. No need to go to Triumph for one.
Here's its location, more about it here:
Here's a thread with the same problem, caused by water in the relay: