Unless you did something dumb, like disconnecting the tach lead from the coil's points terminal on the coil and replacing it on the (+) terminal, I suspect that the new alternator is putting voltage spikes on the power line. That's a concern, as a modern alternator's output should be pretty clean. I agree, it might be a good idea to take the alternator somewhere to check it.
If the alternator is OK, and you can find someone with an oscilloscope, it would be a two-minute task to confirm the spikes. Then adding a filter capacitor from the tach's power lead to ground should fix it. I would use about 100 microfarads, with a breakdown of 25 volts or more. It will be polarized, so the (+) terminal must be connected to the power lead and (-) to ground.