An Axis-aligned French Navy in the Mediterranean

Could Britain hold Gibraltar and use the Mediterranean if the French Navy joined the Axis?

Probably, but at much higher cost, while French-Italian command friction and political mistrust would complicate the campaign.

Britain would probably retain Gibraltar in this counterfactual, while Mediterranean operations would require considerably more force. Control of the whole sea is a different proposition: for much of the actual war, neither side held it exclusively. A French fleet fighting alongside Italy would also face command disputes and the difficulty of accepting German direction. British mistrust after the armistice is located chiefly in French political assurances, despite continuing professional trust between the navies. That distinction helps explain the reasoning behind Mers-el-Kebir and Force H. A further uncertainty is whether French crews ordered to fight for the Axis would obey, with ships potentially going to sea and failing to return.