I've got a fault with the RH rear indicator. When the RH indicator is turned on, the rear bulb doesn't illuminate fully (like a dim glow), and the RH and LH reversing lights also glow dimly. It looks like there is a short between the reversing lights and the RH rear indicator bulb somewhere.

There's no problem with the LH indicator.

Also the RH reversing light doesn't work. I suspect this is connected with the same fault.

I've got a multimeter and the JTIS CD.

Any idea where to start? Is this a common fault?

Well you are prepared to troubleshoot so I would just start.
No it's not common at all.
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I would start at the housing assembly. I had water leaking into my trunk (common problem) from the tail light seals. If some water is getting into your housing from the seal, it could wreak havoc with the light bulbs and their wiring.

Just been outside and removed both connections from the light clusters and my reversing beepers came on and reverse gear was not engaged. Is this normal or could the reversing aid be the problem?

I know this sounds weird and dumb but look for blown fuse in rear fuse box. had multiple cars come in with same concern and end up being fuse.