Almarg's choice seems to want to get rid of the risk, of having these two amps outputs, that may be tied together in the speaker some way. If this is happening, it could be a recipe for disaster. Checking with the meter should answer that easy enough.
Another possibility I'm thinking, is mechanical feedback. This would be similar to a guitar, or a microphone picking up the vibration from the speakers, and causing that feedback. The same happens with records. I've had this problem over the years (all size rooms) with vinyl. If the speakers are close enough to the speakers, the record will get the vibration. The result, feedback. I remember this happening is a big Disco, and it sounded like that large building was going to fall apart, until they turned the volume down.
So, the OP has a lot more bass now (higher power solid state), and most likely less treble and midrange (low powered tube), with this new mis-matched level combination. If you don't hear enough highs and mids, you turn it up, and also receive more bass. This is a possibility that I'm thinking might be happening. Try a CD at your risk, to see if this is the problem.
Then another possibility. I'm assuming both of these amps are using feedback. This is taken off at the speaker output area. This output is signal is then fed back into the front end (input area) of an amp.
If either of these amps is doing something a little strange (odd), it may handle that (odd signal) in its own feedback loop, and play fine by itself.
Now if one of these amps is doing something a little odd itself, and that odd signal (that doesn't bother the amp with it) is being fed right back to its input tied right to the RCA area. And, bonding this amp with a different one may give that little odd (feedback) signal to the other amp, that its (RCA) inputs are tied to.
So, if this is happening, that little odd signal in the one amp (that doesn't mind it), can be driving the other crazy, so to speak. And, these two amps may not ever work together, if one these amps is wired this way, and being the problem.