oneclick 

A while ago I got given a couple of rolls of fabric tape for use on rims. It's some sort of synthetic, strong and not very stretchy, mildly sticky on one side. The numbers written on the inside are widths in mm:

verktyg 

Hockey stick tape? I bought a large roll like that for a 1960's bike that has a smoother finish on the original bar tape than standard cloth bar tape.

verktyg

dedhed

Likely gaffers tape

curbtender

Try silicone tape. It sticks to itself without gumming up your rim.

nikku 

seconded. Gaffer’s tape is great stuff btw.

unworthy1

could be Gaffer's (or Gaff as some call it) tape but the tell will be the adhesive.
Real Gaffer's tape has a special synthetic rubber based adhesive that's usually off-white color. and the reason this tape costs so damn much!
But what's odd with the examples shown is they are cut to metric widths, all the stuff used in NorAm film/TV industry is cut to Imperial inch-based increments and very very rare to find it narrower than 3/4", also this stuff has a slightly fine weave to my eye but that could be just the pix and lighting.
If it was custom-cut Gaff tape then it's bound to have cost even MORE than the stock 2" or 3" wide stuff they go thru by the case in Hollywood.

clubman 

I've worked as a gaffer's assistant and never seen anything like it. Off-white and gray were the two colours we used.

oneclick 

The adhesive is either clear or black, it's thin, and not particularly sticky. The tape itself is very thin, not a hockey-stick tape in material or dimensions or adhesive.

It works great as rim tape and it's so thin that a roll of it has done a lot of wheels.

eom 

Spike tape?
https://en.wikipedia.org/wiki/Spike_(stagecraft)

Mr. Spadoni 

Binding tape used to stabilized heavy materials before sewing, like hockey boots, shoulder pads, etc. ? It’s non stretchy,
has little adhesive comes on a wide roll and is cut as needed to specific widths by the folks doing the assembly of the finished products.

John E

Never underestimate the importance of a properly installed rim tape or strip. My mountain bike rims take 10mm rim tape, which fits nicely in the channel where the recessed spoke heads lie. Even though there was no direct contact with the nipples or spoke ends, I started getting punctures on the inside surfaces of my innertubes, presumably because the rubber would protrude into one or more spoke wells. At first I thought I had been accidentally molesting the tubes with my tire levers, but I belatedly figured out that my decades-old technique was fine and that displacement and loss of adhesion in the rim tape was the culprit. I installed new tapes on both rims two months ago, and so far no more punctures.

oneclick 

Brand name Varia, 13, 16, and 20mm.

The weave and strength are such that it can make a really good valve-hole. I overlap about 4 spokes-holes each way, than a blunt tapered point carefully worked will push apart the threads.

Mickey2

I have seen gaffer tape (or any other brand name) used as rim tape. Coroplast is another brand that probably has provided polyester, achrylic and various technical tapes for two or three generations back. On a 1969 Raleigh I have the original rim tape was a non stretch unbleached cotton tape (binding tape the same as for clothes making) and the ends was sort of joined together with metal hooks, no glue used. These were woven in a fish bone pattern. I think you still can get the Velox version of this with sticky backside. I don't know about the Varia brand, and you found what you searched for, but if you are looking for something similar, any woven nylon adesive tape could be an alternative?