Mediterranean product is of ultimately Greek character (4)
I believe the answer is:
feta
'mediterranean product' is the definition.
(I know that feta is a type of Greek cheese)
'of ultimately greek character' is the wordplay.
'ultimately' says to take the final letters.
'greek character' becomes 'eta' (letter from the Greek alphabet).
The last letter of 'of' is 'f'.
'f'+'eta'='FETA'
'is' is the link.
(Other definitions for feta that I've seen before include "Greek 8 across" , "Salty Greek cheese" , "Goat's or sheep's milk cheese" , "10 down cheese" , "White Greek cheese" .)