Stivervii, the JJ KT77 are NOT compatible with the Consonance Cyber 800 tube monoblocks. And, despite what some try to tell you, neither are 6550 and KT88. This is the price paid for the convenience of this cathode bias (autobias) circuit.
After CES2008, the owner of Consonance stayed with me for almost a week, and was astounded to find my assertion correct, as he was sure the amps could run the other tubes. Believe me, I'm a tube roller, so I've already tried as many combinations as I felt worthy of trying. He felt the reason for it is perhaps in the way the amps behave on 110/120V AC. We may address this going forward, though due to other projects having priority, it will not happen soon. If your friend's amps truly does use them, he must certainly have a pair with different values in the bias resistor positions. Otherwise, it's July 4th any day of the year.
Unfortunately, Mariusz, NOS Mullard tubes command a prince's ransome - Mullard KT77 being a king's ransome. A few of us in my audio group got together over a period of several sessions (using different amplifiers - Jadis, Granite, Quicksilver, Consonance) and performed a tube shootout between 1960's vintage Mullard EL34 XF1, JJ Blue Glass E34L, JJ Clear Glass KT77, and JJ Blue Glass KT88. Another party also included the EH EL34. The intention was to determine whether any current production tube could displace the Holy Grail that the Mullards have come to be considered.
The winner was the Mullards. They performed best from top to bottom. The JJ E34L were voted best when price was factored into the equation, and some folks rated their midrange best in a few of our scenarios. I don't want to give anyone the idea that the JJ E34L sounded like the Mullards; they did not. The weak point of the JJ EL34 was the definite lack of extension and power in the bass, for the KT77 it was the recessed midrange (though, some also didn't like the overly fat mid/upper bass, which they felt out of balance with the rest of the musical spectrum), and for the KT88 was every area other than output power.
But, I agree with Nickword's suggestion of trying JJ E34L tubes to get where you are trying to go, Mariusz. Though the JJ E34L is my favorite current production output tube in audio, as things are with the stock amps, switching the EH 6CA7 out for them does not yield improvement, in my opinion. The very slight gain in the mids is more than offset by the lossess in overall power and the low frequencies - extension, impact, and definition. Your amps being modified by Bob Backert however, may behave quite differently in this regard than the stock products, and produce the results you are looking for.
One note of caution, however, is that the reliability of JJ output in general is eye opening. Because of their sound, I was pushing Consonance to use them instead of the EH (apart from the 6CA7 in the Cyber 800) tubes used in most of the products. Testing proved that this was definitely not the way to go (I don't want to get into the numbers in terms of failure rates here), though the Consonance Cyber 100 Signature KT88 15th Anniversary integrated is using them. If one of your amps encounter a bad output tube, it will definitely go down, and you'll be taking it in to either Bob or me for service. I've had amps in for this repair, and in both incidents, the output tube the customer used was the JJ E34L.
DISCLAIMER: I am the importer/distributor for all Opera Audio/Consonance products for the USA and Canada