The most common and most likely answer to this crossword clue is the 10 plus 2 plus 7 letter word DIFFERENCE OF OPINION
I've seen this crossword clue in The Times
D
I
F
F
E
R
E
N
C
E
O
F
O
P
I
N
I
O
N
Buddy explains!
The straight part
We know that "which has some falling out" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case which has some falling out = difference of opinion.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.