It could be Broussonettia papyrifera. In which case they are not individual plants but suckers joined by a network of underground connections. That’s why pulling individuals has no long term effect.

thanks for your help! Mine grew to 4 feet before we had a chance to pull them down and none of those stalks had any fruit, berries, or flowers. My adjacent neighbor gave up and let her weeds grow over 6 feet tall and they didn't have berries either. However, some of those tall plants have bark identical to the bark of a broussonettia papyrifera. Either way, I'm guessing this is going to be quite a challenge to get rid of. No wonder all of my other neighbors have poured concrete in their backyards.

The lack of fruit indicates nothing. These are large shrubs or small trees, much taller than 6 feet, and if yours are dying back every winter they are unlikely ever to get to flowering size. Also they have separate male and female plants and the males, obviously, never fruit. If they are suckers they will all be coming from one parent and will be genetically identical to it. So they would all be the same sex. Further information. Note the statement: Paper mulberry is a vigorous pioneer species, which can rapidly colonise forest clearings and abandoned farmland.

I can't be certain that this is what you have from just one photo taken at a distance but it does appear to fit the characteristics you describe.