It's my understanding the two coexisted at one point - what was the relationship between them both? Did the Byzantines view the Holy Roman Empire as a legitimate establishment, or did they see it as a "pretender"? Like Voltaire pointed out, it was neither holy, nor roman, nor an empire.
Lots of room for more input, but you may be interested in some earlier posts
What kind of relations did the Byzantine Empire and the Holy Roman Empire have?
What did the Byzantine empire think about the Holy Roman Empire?
How did the Byzantines view the Holy Roman Empire and vice versa?
all of these posts have been archived, so if you have follow-up questions for any of the users, just ask them here and mention the username to notify them
Here are my two posts on this issue: