Unofficially switched PhD supervisor to one at a different institution. Can I get this officially acknowledged? 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?
That is a question that only institution B can answer, but I doubt that they would be agreeable due to your formal link to A. 

What you might be able to achieve is to have supervisor B included in some more or less official way in your process, though, as you say, he/she already declined. 

So, supervisor B doesn't want to be your examiner, but perhaps, subject to supervisor A's sense of things, B could be a co-advisor and also sign your dissertation, giving titles and institution. A may even find it advantageous to have an association with B. 

But that is largely a personality issue. I doubt that either institution would object.