I was discussing automatic tap changers are you saying they are or are not per phase or switch as a 3-phase unit?
They do have load tap changers that are part of the three-phase transformers (or added on). Normally these would shift all three phases together. Some of these are made of single-phase units and have individual selector switches so I guess you could build them to step separately if you had a control panel that allowed that. Even for the normal case of stepping together, individual units can fail and cause an unbalance. The problem with these type units is that you have to take the whole thing out of service for repair.
What we normally have is three separate single-phase units that operate independently. This gives more operational flexibility and makes repairs easier. These do not step together but I suppose you could force them to step together by having the separate control panels talk to each other. However, I think this would remove some of the benefit of having separate units.
I do see value in having the units talk to each other to control unbalance but that has not been the way they have historically operated as you normally use the same set points for each control panel and they stay in balance. When one control panel goes haywire, the cross-communication might fail as well but there may be a possibility that the other two units could react. But what if the bad unit runs away to the max: do you really want the other two to follow along?