there's a dash mounted tach available for Volvos (you might find them in Australia). It lets you retain your ribbon type instrument panel instead of changing it to the Rallye type cluster, and at the same time have a tach. I saw that on a 72 142 I was "working on" a couple of weeks ago.
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and the bricks keep on rolling....

Oh, silly me. That will teach me not to reply at the end of the day just before going home. Your tach does return to zero when off, so it is not sticking.

Then it would be the internal circuitry. If you are electronically inclined then you would know what an electrolitic capacitor is. They are the first components to age, and replacing them will possibly fix the problem. Alternatively, the solder has cracked due to age and all the components are not connected any more.

Cracked solder is very common with newer Volvos (a Bosch problem) due to the manunufacturing processes. On the 240 and 740 it shows up as a failed automatic overdrive relay or fuel pump relay.

Have fun...

This is the OEM VDO tach and when everything is off, it reads zero.
When the ignition is turned on (lately) it goes to 1000 rpm. At idle
it reads maybe 12-1500 which is faster than it has been idling.

Before, it did like it was supposed to, reading zero when the engine was
stopped (with ignition turned on), pulsing a bit with cranking and then
reading 800 at idle and higher during operation, apparently correct.

Now, with ignition off, it reads zero. With ignition on, even with the
wire to the distributor disconnected (or connected) it reads 1000.
As I mentioned before the tach in my 73 164 did the same thing. Gotta be
something in the power supply or internal circuitry if it does it with no
signal input.