The most common and most likely answer to this crossword clue is the 7 letter word DIARCHY
I've seen this crossword clue in The Guardian
D
I
A
R
C
H
Y
Buddy explains!
The straight part
We know that "government style" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case government style = diarchy.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 DIARCHY mean?
DIARCHY noun- Rule by two people.
- A state under the rule of two people.