Quit after Conservative split (6)
I believe the answer is:
cleave
'split' is the definition.
(synonyms)
'quit after conservative' is the wordplay.
'quit' becomes 'leave' (synonyms).
'after' means one lot of letters go next to another.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'leave' after 'c' is 'CLEAVE'.
(Other definitions for cleave that I've seen before include "Chop or split" , "remain attached" , "Split apart - stick together" , "Part" , "Split with the grain" .)