Two integrated amplifiers, tube and solid-state connected to same speakers

You need to manually connect and disconnect the speaker cables, that is the safest method. That’s my ideal setup, one high quality tube integrated and one high quality solid state integrated. SS will always be on, connect the tube when I’m in the mood.  I have the tube integrated now, in the market for the SS.

If you have pre-amp outputs one of the solid state amps you can run that to another amp.  At that point you only need a power amp, not another integrated.
The hard part is connecting to just one pair of speakers.  If you accidentally run both amps at once you just might blow something up (probably the tube amp).  Niles makes/made an amplifier selector to all for 2 amps and 1 pair of speakers.

There is not a separate power switch to shut off the power amp section on an integrated amp. The amp section will still be active when you use it!