The province of a genuine Frenchman (5)
I believe the answer is:
realm
'the province' is the definition.
(thesaurus)
'a genuine frenchman' is the wordplay.
'a genuine' becomes 'real' (I've seen this before).
'frenchman' becomes 'm' (abbreviation for Monsieur - like 'Mr').
'real'+'m'='REALM'
'of' acts as a link.
(Other definitions for realm that I've seen before include "Dominion" , "Country" , "Monarchy" , "Area of interest/activity" , "field of study" .)