Jennifer Lopez and Marc Anthony Now Getting Along

Sources very close to the former couple tell TMZ ... months before the breakup, there was virtual radio silence between the two. We're told when they did talk, it was loud and contentious ... often ending in tears on Jennifer's part.

But since the split, we're told their bond has materially strengthened. They are communicating and co-parenting well and not fighting at all.

We're told they're actually happy to do joint appearances for their new show, as well as charity work.