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If your tape is making a mess you really have some problems. The most worrying is that the tape may be good quality but something in your kit is attacking it. Even good tape can't abide being exposed to gasoline fumes or oils.

The worse stuff is 100% DEET insect repellent because it is persistent and spreads. It is oily and attacks some plastics. A bottle of insect repellent can destroy a good portion of a kit. Messing with a roll of tape is the least of your problems. A small bottle of repellent got lose in a pack and it caused the plastic base of a compass to fog up and crack and it dissolved part of a handle on a SAK. I am more careful with that stuff now.

If the tape isn't being attacked then you still have two problems. First, the cheap tape is making a bothersome mess.

Second, your hauling around cheap tape that is likely to be useless when you need it. You would be better off leaving the tape behind. Less weight, bulk, mess and, you would know you don't have tape. Better to know what you don't have than to feel confident in what you have and find out it is useless.