The most common and most likely answer to this crossword clue is the 9 letter word ANTIPOPES
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Buddy explains!
The straight part
We know that "some old rival church leaders" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case some old rival church leaders = antipopes.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ANTIPOPES mean?
ANTIPOPES noun- A person who claims or claimed to be the pope, usually as the result of a disputed election or deposition, but is not considered by the Roman Catholic Church to be the real pope.