I’ve been to a few (and they would refer to themselves as being “Western Orthodox” or “Western Orthodox Catholic” )
The Antiochian Orthodox Church has a Western Rite and there is also one in the ROCOR (they have a Western Rite Benedictine Monastery in Hamilton with the Rev. Fr. Abbot James Dechenes - a very holy man).
There are actually two liturgical Western rites - one called the Rite of St Gregory for former Roman Catholics and the other the Rite of St Tikhon (patriarch of Moscow) for former Anglicans.
I can tell you that they do NOT commemorate the Bishop of Rome, but their respective local Bishop and also their Patriarch.
Lancelot Andrewes Press publishes their books, including the Book of Common Prayer (according to Orthodox Catholic usage).
The Anglican heritage appears much more fully represented within Orthodoxy than within RCism. It is a magnificent, inspiring heritage!
Alex