My wife is highly allergic to PI. She will get swollen areas covered with layer upon layer of blistered and using sores, several inches across, turning the skin dark purple all around it. It can take weeks for the oozing to stop sometimes. It goes so deep into the skin it causes scars that last for over a year. Because of this, and because she loves to work outside and to landscape, we're always on the lookout for PI. PhysAssist covered the treatments well. We use Tecnu religiously, and found it works very well if you follow the directions, and then some. You need to rub it on/in your skin, and continue to massage the affected area for several minutes. And I mean by the clock, at least 3 minutes. Add some more if it starts to dry out as you rub it. The best way I can describe it's action is to think of it removing the oils just like Goop or DL cleaners remove dry paint on your skin. You have to give it several minutes to break down the paint/oil, and then all of a sudden it dissolves it quickly. I apply the Tecnu even to areas that have already blistered, and it really slows the reaction and speeds healing.
As others have said, I chop the vines with loppers or a machete, and paint the stump. I use Crossbow (or generic Cross Roads) painted full strength on stumps, and also spray lanes with that using 2 oz per gallon of water to kill PI quickly without harming grasses.