The most common and most likely answer to this crossword clue is the 13 letter word PRIEST-RIDDEN
P
R
I
E
S
T
-
R
I
D
D
E
N
More clues leading to the result PRIEST-RIDDEN
Buddy explains!
The straight part
We know that "dominated by ministers" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dominated by ministers = priest-ridden.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.