1st year, put down weed barrier under 24" of loam, compost, manure mix. Nothing grew more than 4-6". In fall dug it up. Oriental Bittersweet vines had grown through the weed barrier and formed a dense, almost impenetrable layer of what looked like thin, brown, stringy seaweed in a clump throughout the entire
garden. 2nd year, after digging up the soil, which was very difficult with the clump of bittersweet vines, I made the boneheaded move of putting a polyethylene tarp under the soil, completely thoughtless as to drainage. So, no bittersweet, nut no productive plants either. Kind of hard to grow in the swimming pool I had made!
Going into the 3rd year I am digging it up again, for the last time, and looking for some type of super strong weed barrier that will keep the bittersweet out but is also water permeable. Any suggestions would be greatly appreciated. At 66 I'm really getting too old for this.Thank you

Your story is another of SO MANY tales of woe caused by landscape fabric. As you've seen, it's not effective as a weed barrier against weeds that will just creep under it. To use this area without just cultivating bittersweet along with your desired plants, the bittersweet would need to be removed. Anything that will allow water to pass through will also allow roots to penetrate. A thick layer of mulch is the usual best defense against unwanted sprouts that occur from above, but you've got a root problem within the soil that won't be deterred by mulch.

A raised bed might be an option, which also offers the added bonus of no longer needing to bend or stoop.
Or some big pots. Are you able to add a pic of this area? Are you trying to have an ornamental or edible garden? Are you trying to get rid of the bittersweet?