Wary bishop is hosting Democrat Eddy near Sicily (9)
I believe the answer is:
Here is my best explanation:
'eddy near sicily' is the definition.
(whirlpool sea monster in Greek myth)
'wary bishop is hosting democrat' is the subsidiary indication.
'wary' becomes 'chary' ('chary' can be a synonym of 'wary').
'bishop' becomes 'b' (abbreviation used in chess).
'hosting' is an insertion indicator.
'democrat' becomes 'd'.
'bis' going around 'd' is 'bdis'.
(Other definitions for charybdis that I've seen before include "Scylla's counterpart", "monster", "Whirlpool opposite Scylla", "Whirlpool, across from 14", "Mythical whirlpool" and "Monster".)