As far as I know there are no Non-Chalcedonian churches (those that rejected that Council), like the Copts and Armenians, that have come into union with the Orthodox while retaining their Rite and praxis; whereas there are a number of these churches that have come into union with the Catholic Church after accepting that Council's decrees and papal authority, albeit, sometimes under questionable circumstances. They do retain their own rites and praxis (often Latinized), as those of the Byzantine rite (Ukrainian, Melkite etc.).
There are, however, some Orthodox parishes here in the US and perhaps abroad that use their own "Western" rite; but there are no Latin rite Orthodox eparchies as far as I know.