How to deal with different stylistic advice from two professors/institutions
The first thing you need to do is determine if there is an underlying issue with your writing that A and B are trying to get at. If there is, fix that problem and the style issues may be ignored. If it truly is a style thing, you need to talk to them and explain the issue. If one supervisor demands that you never use the passive voice and instead write in the 1st/3rd person and the other supervisor demands that you never use 1st/3rd person and prefers everything in the passive voice, then somebody is going to have to give.

Once you determine it is purely stylistic, you need to decide which style you want to write in. This should probably be influenced by the recommended style of whatever journal/publisher you are targeting and/or any archaic rules your department has about theses. Then approach whoever you are agreeing with, get them on your side and finally explain to the other person the approach you are going to take. They might have a good counter argument or they might give up.