If it continues to run key in or out until a fuse is pulled as you say, you most definitely have a short to ground in the ignition switch or circuit. Whatever is shorted to ground is tricking the ecu into being in run mode and keeping things going until it sees the circuit that fuse controls go dead when the fuse comes out.
Start in the ignition switch as they are known to be faulty. I am assuming you ride classic or standard. Triumph thought it a good idea to switch absolutely every circuit in the bike through a plastic switch, so it gets hot and warps or melts. After that, you are looking at the joy of troubleshooting connectors and the loom.