They indeed don't need that. Nor PoS. What is more, when there is a "consortium of companies that do no trust eachother using a private blockchain" they imply

They indeed don't need that. Nor PoS. What is more, when there is a "consortium of companies that do no trust eachother using a private blockchain" they imply three things: * They have a permissioned system: not everyone can participate. This means there is a mechanism to punish adversaries right there: just remove their permissions! A large part of what "a blockchain" is all about, is dealing with sybil attacks stemming from the requirement of it being permissionless. * They know eachother. Hence there won't be a sybil attack possible. * They have established communication channels (otherwise it is impossible to grant access to the private blockchain). Hence the Byzantine problem is non-existing or not an actual problem. Maybe, very maybe, there is a case for a private blockchain in which there are many participants and a system to fully anonymize all these participants once they "are inside the firewall". I really cannot comprehend this, though. But that would make a case for a private blockchain, becayse sybil attacks are then again possible and need to be dealt with.