In the UK / G-reg regime, any flight with a proper instructor can be logged by the LHS as PU/T, even if the LHS is capable of being legally PIC.
You can take your instructor to Mongolia and log the whole lot as PU/T.
If the flight is outside your license privileges (e.g. your papers have lapsed) then you must log it at PU/T, and the RHS must be a proper instructor.
If the flight is outside your license privileges (e.g. your papers have lapsed) and the RHS is not an instructor, then you cannot log anything.
FAA rules are different. The tendency there is to log everything as PIC, which is technically right even for an ab initio PPL student since he is PIC under the privileges of the US Student Pilot Certificate.