Not married - who would inherit our french property?

If one or the other of you was to die then that person’s children would, under French inheritance law, inherit part of that person’s share in the house, according to the French inheritance rules (eg. half if a single child, two-thirds for two children etc).

A will would be needed to make sure the remaining part went to the surviving partner.

Buying with a tontine clause is one option if you both want the other partner to inherit the house.

You could also make use of the new EU inheritance regulation to opt for UK inheritance rules, so as to leave everything to the other person in a will.

Note though that, in relation to inheritance tax, it would be considerably better to marry or take a French civil partnership (pacs) to avoid heavy levies (60%) for the survivor.