Is there a semipermanent -- that is, removable -- adhesive tape that maintains its adhesion under water? Or is there a tape used in surgery or medicine that maintains adhesion while being impervious to water or perspiration?

It depends on what you want to do. For something like fixing a leak in a boat, the best emergency stuff IMHO is the aluminum tape made for furnace ducts, it comes in a roll but it has a peel and stick backing.

Gorilla Tape is the toughest of the duck tapes I have found, but I haven't found a tape yet that can be stuck on a wet surface.

The only thing I have found to repair wet surfaces is an epoxy adhesive putty made by a company called "Fiberglass Coatings" in florida, they have a website. You can mix up a bunch of putty, put it on a cloth pad, and it will stick to whatever you smoosh it on.

As Mark insinuated...

It would be easier to help if we knew what you were fixing.

I am in the pressure sensative tape business, but never heard of such.

If there is, I would be very interested

The 3M fiberglassed reinforced tape will hold lobster claws closed for two weeks underwater, when wrapped once around with the adhesive side sticking the back of the tape. This is according to some study done in Halifax in the early 80's, I haven't tried it myself. Perhaps your local lobster salesman can help.