The most common and most likely answer to this crossword clue is the 4 plus 6 plus 9 letter word GOOD FRIDAY AGREEMENT
I've seen this crossword clue in The Daily Telegraph
G
O
O
D
F
R
I
D
A
Y
A
G
R
E
E
M
E
N
T
Buddy explains!
The straight part
We know that "conciliatory political initiative" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case conciliatory political initiative = good friday agreement.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.