Originally posted by The_Tempest
What authority does he have?Sure we do: because he doesn't actually have any official power to do anything beyond what he is charged to do with the Emperor.
...why would Tarkin be "superior" on the Death Star?
His authority is that he is the Palpatine's right hand man and enforcer. We see him command and take the initiative several times to serve the Emperor throughout ANH and Rebels. He is not going to consult with Sheev on every action he is going to take. To bring it back to R1, if Vader thought Krennic was more fit to defend the DS than Tarkin presumably he could give Krennic the command. Now he can't just do this because he likes Krennic better rather it must best serve the Emperor's own interests.
Ultimately, nothing in the film nor the novelization suggests Vader has consulted with Sheev prior to his conversation with Krennic. It appears that these are Vader's commands not Palpatine's direct commands. Whereas in RotJ he makes it clear that he is expresssing the Emperor's commands.
Tarkin would be his "superior" as he commands the DS. It's akin to a General deferring to the captain of a naval vessel.