I've never tried using auctex remotely, but in general if you have a file open in Tramp, you can also run arbitrary shell commands with M-! and M-& remotely, as

I've never tried using auctex remotely, but in general if you have a file open in Tramp, you can also run arbitrary shell commands with M-! and M-& remotely, as well as opening shell buffers that let you run commands remotely as normal.