Revealing mistake: When the Doctor sabotages the inertia neutralizer, the wall wobbles.
Revealing mistake: The TARDIS manual Romana shows the Doctor quite obviously has a couple of hand-written pages jammed in so that they can be easily ripped out.
Revealing mistake: There are times in this episode where the strings holding up the Captain's trophies are visible. Additionally, the "billions of tons" of planet sways back and forth slightly.
Revealing mistake: The queen, suspended in the last few moments of life, can be seen to breathe rapidly and blink.
Revealing mistake: The split screen used to make two Doctors appear simultaneously is apparent when the Captain's chair appears to blur while both are in shot.
Revealing mistake: When Pralix lifts the spanner in the composite shot, bits of debris lift up with the spanner.
Revealing mistake: K-9 has a bit of difficulty in the mansion in episode two. The thread pulling him is visible and Baker has to help him over a flagstone.
Revealing mistake: Zadek claims the water door into Castle Gracht is a foot thick. But it is obvious when the Doctor crawls through that it is paper thin.
Revealing mistake: When K-9 is cutting through the wall of the hut, the cut overtakes his laser at one point.
Revealing mistake: The footage of Kroll eating Harg is obviously reversed, as evidenced by the fact that the smoke from the pipe is heading into it, whereas in every other shot it's flowing out of it. (00:23:00)
Revealing mistake: One of the stones wobbles when a crow lands on it in episode one.
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 ★