Caribbean island's medical officer terribly stern with rogue (10)
I believe the answer is:
montserrat
'caribbean island's' is the definition.
(Montserrat is a Caribbean island)
'medical officer terribly stern with rogue' is the wordplay.
'medical officer' becomes 'MO' (abbreviation).
'terribly' indicates an anagram.
'with' means one lot of letters go next to another.
'rogue' becomes 'rat' (rat can mean a nasty person).
'stern' is an anagram of 'ntser'.
'mo'+'ntser'+'rat'='MONTSERRAT'
(Other definitions for montserrat that I've seen before include "somewhere within WI" , "Country" , "One of the Leeward Is." .)