I just had a look inside a spare cluster. I have a Bertone anticlockwise tachometer. The tacho driver circuit has an 8 pin DIL IC, with the number 8610 on it. This kind of suggests something like an LM8610 IC, but I cannot find a record of such an IC. Anyway, it looks like a simple op amp based integrator circuit, with a few cheap resistors and capacitors.
To get to this circuit board, you will need to dismantle the tachometer, which carries a good risk of ruining it anyway!
If you can identify the input and output of the driver circuit, you can check its operation by applying a signal from your coil (either use a signal generator or connect it to your engine). With + and - connected, you should be able to read an output voltage with your test meter, which should increase smoothly with increasing engine revs.
Similarly you can check the moving coil meter by applying +, - and a variable voltage to see if it responds.
It may help to look at some application circuits for ICs like LM2917, you could make up your own tacho driver circuit based on them.
I don't really have the expertise to tinker with this sort of stuff, I would be inclined to source a s/h replacement tacho or take it to someone like Speedy Cables!