Swap it for black tape.
Often, a lot of the blackness is finely powdered aluminium that's been worn off the rim by the brakes and got onto your hands last time you had a puncture. Solvents and detergents don't have much effect on it, and it works into the texture of the tape where scrubbing doesn't get at it easily.
If you persist with white tape, I recommend adding a pair of latex gloves to your puncture kit.