Continuity mistake: The amount of dirt on the Doctor's face changes between the Bath House and the exterior filming.
Continuity mistake: When the Doctor enters the desert room in addition to his coat, he is wearing a brown waistcoat with a white shirt underneath. When the shot cuts away and the back to the Doctor, we see him take off his coat but he is no longer wearing the waistcoat. Later when he collects his coat after the acid bath scene and he starts to look through the wardrobe for a disguise he is mysteriously wearing the waistcoat again.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★