My PhD (I have 1 year left in it) started out at institution A with supervisor A. For reasons not worth going into, I am now, in most practical senses, being supervised by supervisor B at institution B. It is the case, for example, that supervisor A asked if supervisor B could be the external examiner of my thesis, but supervisor B declined because they felt they were "much too close to the project".
I once asked to move my PhD entirely to institution B, but was told my funding was attached to institution A. This wasn't the end of the world; the arrangement we have is working pretty okish, and there's nothing acrimonious going on. So I am fine not changing anything practically
However: institution B is, frankly, much more prestigious than institution A. When I finish my PhD I'd like to be able to put institution B on my CV. I am thinking of doing some private tutoring after I get my PhD and the difference could genuinely be an extra 10% of my pay.
Can I get something official associating my PhD with institution B?