Dear RonF,
"Whereas the Anglican Rite is part of the Roman Catholic Church, under authority of the Pope,..."
No, there is no Anglican Rite which is in communion with the Bishop of Rome. There are Anglican use parishes within the Latin Rite.
The folks at this link:
http://www.anglican-catholic.com/ed.htm
which NYer asked about, give no appearance of communion with the Bishop of Rome, and I don't believe that they claim any. They are just like the Anglican Catholic Church, from a Catholic perspective - another break-away group from the Anglican/Episcopal communion that is unhappy with the direction of that communion.
sitetest
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.