The most common and most likely answer to this crossword clue is the 9 plus 9 letter word COALITION AGREEMENT
I've seen this crossword clue in Independent
C
O
A
L
I
T
I
O
N
A
G
R
E
E
M
E
N
T
Buddy explains!
The straight part
We know that "demise of LibDems thanks to this" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case demise of libdems thanks to this = coalition 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.