O.K. This is, truly, my mistake. What I was thinking of was "Anglican Use", not "Anglican Rite". According to
this site, there are 6 or so parishes in the U.S. that are as I described, using a liturgy similar to an Anglican one under the authority of the Pope. Check it out. And, my apologies. I spoke without making sure I knew what I was talking about. I generally try not to do that.
No big deal. Easy mistake to make.