Tube integrated with real bass. Does it exist?

If Hybrids are OK with you then the Pathos TT, providing you maintain a speaker with similar specs to those you mention.

If there's a real bass in class D amps(fact!) than there's no real bass in tube integrated or what is real bass after all?

Ayon Triton is a great suggestion for tube amps with speed and bass.

There are many others but typically tube amps which try to do too well in bass lacks the magic in the mids and highs. Many examples exist. So you probably need to find a good tube which can drive your speakers in a reasonably sized room. Bass speed is the criteria to look for, not exactly how low it goes. 60 watts push-pull should be a good starting point.

Many suggestions.
I asked about only one aspect of performance, but of course there are many others. If an amp or any other piece in the chain doesn't do rhythm and dynamics well I don't listen further, no matter how otherwise glorious the sound might be. I equally dislike sleepy and hectic sound. This just has to be right. And any good tube amp will do certain things well, so I wouldn't worry about it much, though some will do them better than the others.
I took a look at the websites of the companies mentioned here. If I was ready for an upgrade now, wanted to keep my new amp for ten years or longer and had no opportunity to audition any of them, I would take a guess and get VAC Sigma, $5000 used. NOS tubes would be next upgrade. In any case, I'd rather wait two/three more years and get a great piece.
Jadis looks like Cartier, a royalty. Question is can it rock and do jazz/rock fusion? Or is it fast enough to follow McLaughlin's guitar? Or able to reproduce the raw energy and at the same time sophistication of El Camaron's flamenco singing? I would have similar reservations about Italian amps. German equipment I have no idea of, never thought of actually. Air Tight looks very esoteric, very elite, with some music would probably be the best. As I understand, Rogue is a great bargain, but as any bargain it most likely can't compete above certain level. Talking used pieces, $1500 for Rogue vs $5000 for VAC, $3500 difference for ten years? There is no difference, if you wait a little longer.
German Octave did catch my attention too, not sure why. In reality, if you want to do it right you have to audition them all in your system, of course. And of course it is not very realistic unless you are prepared to buy them all one by one and then sell them all but one.

Inna - Just remember you can change the overall sound of a tube amp by swapping tubes. I never heard any 2 different manufacturers or types sound the same. They all have their own sonic signature. Example no 2 KT88's - EL34 or 6550 sound the same from different manufactures. Same holds true for little tubes like 12ax7 - 12au7 - 12at7 etc.

That said a lot will depend on your speakers. When I had Dyn C1's I preferred a warmer sounding tube like the stock SED KT88 or vintage Tung Sol 6550 solid plates. But when I went to Raidho D1's those tubes sound mushy and dull. I then preferred a more linear sound like a vintage KT88 SED 6550 or Ei KT90. Didn't care for the KT120's with either speaker.

That said I give the Octave my highest recommendation. Extremely reliable - easy to bias should you want to swap tubes and great protection for the output tubes. Not to mention soft start which extends tube life and auto off when no music is detected after about 7-8 min. As quiet as the Octave is you can add the black box or super black box to lower the noise floor even more and gain a bit of headroom.