Tube integrated with real bass. Does it exist?

Yes, Ayon Tritton 3

Depends on how much flexibility you need for the "integrated" part. I had a pair of Air Tight ATM-2 monos, which included two inputs and a volume control - so just enough if you use one digital and one analog source. They certainly had real bass and could drive your proposed speakers with no issues, but would probably run you about $26k today.

I'd put my bet on Modwright's KWI 200 at 200 watts/channel. Their amp's bass is tuneful, punchy, and dynamic. Can get one with phono and with dac built in.

A tube integrated with real bass?? I would suggest looking at something that has KT120 or KT150's as output tubes. I will admit EL34's lack the punch in the lows and the sparkly highs but do have mids to die for. I would consider an Octave V80SE followed by the V110. The V80SE has KT150's as standard output tubes where the V110 uses KT120's. BTW Octave had adjusted their US pricing in the spring due to the low Euro. Yes I owned the Octave V70SE then upgraded to the V110. BTW Octave is built like a tank with great output tube protection and easy biasing.