I agree, one time w/o a wrapping is enough of a lesson.

The wrapping should not degrade too rapidly, but not promote girdling either. I used some paper-like product developed for the purpose. Under that, against rough bark, I used cotton futon stuffing to plug the crevasses. Ants are really able to adapt and find their way around barriers; tall grass, under plantings etc., will bridge past low lying bands. High bands directly on the bark will catch you during every visit for pruning, thinning, harvesting. For years the stuff stays sticky. You may need to replenish the bands when they become overburdened.  Another pro for banding material.

Tanglefoot is also used to cover the Apple Maggot fly red globe/attractant traps. Better minds than mine figured out if you but a disposable baggie on those before you paint them up, you'll be more likely to reuse them next season. I don't think anyone wants to learn what process and solvents one would need to clean off the Tanglefoot. Traps get pretty gross with targeted and non target carcasses. 

Ants mark their trails with formic acid that they exude from their bodies so they can find their way back and forth and let other ants know where the goods are to be found.

You can neutralize the formic acid by washing with baking soda solution which is non toxic to trees trunks (and floors and countertops). This confuses the ants and they will wander around having lost the trail until one of them recreates it later on.