Germans had invaded the Channel Islands on 28th June 1940, and then, handed them over to France. They had little strategic value, but they had symbolic value : the only British territory under Axis occupation. They also served to create bad blood between Britain and France.
On 14th November 1940, the RAF carried a raid against the Renault and Panhart-Levasseur factories, near Paris. However, bad weather led the RAF to accidentally bomb Paris itself, killing thousands. Moreover, a downed British plane crashed on the Eiffel Tower.
France retaliated by launching a terror raid on Plymouth (from Guernesey). This initiated a cycle of terror raids. British attacked Brest, Saint-Malo, Le Havre, Caen, Arras, Lille and Calais repeatedly. French did the same on Plymouth, Exeter, Bornemouth, Portsmouth and Canterbury. There were downed pilots lynched by civilians.
The Regia Aeronautica, eager to prove an equal to the Luftwaffe, adopted the same strategy as the Armée de l'Air.
When British met French and Italians on the ground (Axis invasion of Malta and Gibraltar, British liberation of Channel Islands), fighting was actually more ferocious than against Germans.
France and Britain would see each other as perfidious backstabbers and hate each other for decades. Since they had been friends since 1904, there were of course lots of British-French marriages, children and friendships, leading to many personal tragedies.