You are looking for a very thick and stretchable tape that's UV-resistant and has a water-proof backing. A highly specialized and very expensive stuff. Something that's advertised as an <i>outdoor duct tape</i> or <i>outdoor masking tape</i>. When it comes to tapes, it's really hard for anyone to beat 3M and that's the one I would recommend: 3M® #5959 Outdoor Masking and Stucco Tape. My guess is that you can probably buy a no-brand <i>outdoor duct tape</i> at about 1/3rd of the price but it never occurred to me to skimp on this type of tape. Consequences of it failing in a rain or wind are just too expensive compared to savings on the tape's cost.
Here's what 3M says of it:
- 90 days UV resistant waterproof backing
- 35 lb. per inch of width tensile strength
- 12 mil thickness
- Polyethylene film over a dense cloth scrim and a blend of natural and synthetic rubber adhesive
- Strong backing pulls through stucco, EIFS, and other heavy coatings for one piece removal
- 21% elongation at break
- Use in temperatures up to 200° F
if you can find something with verifiable similar specs, go for it. Otherwise just get the 3M tape for the peace of mind.