FD - I wrote:
"The question which has to be asked is "Why is there an authorised instructor on board?"
If for any reason other than as a passenger, then the authorised instructor must be providing some element of training or supervision. In which case, he/she must be the aircraft commander."
In your scenario, your 'instructor mate' acting as a passenger wouldn't be P-anything, nor could he/she be paid.
But if some pilot wanted both to fly as PIC and have some sort of comfort blanket of taking an instructor along to hold his/her hand, then no, the pilot would be acting under the supervision of the instructor and would not be entitled to log the flight as PIC.