We don't have synonyms for exarch yet — it may be a rare, specialized, newly coined, or misspelled word. The closest thesaurus entries we have are eparch, hexarch and enarch.
(n) :
in the Byzantine Empire, a governor of a distant province; in the Eastern Christian Churches, the deputy of a patriarch, or a bishop who holds authority over other bishops without being a patriarch
(n) :
in the Byzantine Empire, a governor of a distant province; in the Eastern Christian Churches, the deputy of a patriarch, or a bishop who holds authority over other bishops without being a patriarch