Persist with boil, almost completely coarse (3,7)
I believe the answer is:
see through
'persist with' is the definition.
'boil almost completely coarse' is the wordplay.
'boil' becomes 'seethe' (seething is a kind of boiling).
'almost completely' means to remove the last letter.
'coarse' becomes 'rough' ('rough' can be similar in meaning to 'coarse').
'seethe' with its final letter removed is 'seeth'.
'seeth'+'rough'='SEE-THROUGH'
(Other definitions for see through that I've seen before include "Help out the needy" , "sheer?" , "Transparent" , "Rumble" , "Stay to the end" .)