Strange fate, producing a Greek export (4)
I believe the answer is:
feta
'a greek export' is the definition.
'strange fate' is the wordplay.
'strange' indicates anagramming the letters.
'fate' with letters rearranged gives 'FETA'.
'producing' is the link.
(Other definitions for feta that I've seen before include "10 down cheese" , "Greek salad cheese" , "Eastern Mediterranean cheese" , "Greek 8 across" , "White Greek cheese" .)