I was told when I last did one, 20 years ago, that it was necessary,
otherwise one gets into tricky debates about how much compensation the cohabitee is entitled to.

Whereas if the cohabitee purchases a % share in the house, the
agreement can simply state that the homeowner just buys that, and has
to find the money within x weeks.

There is no mortgage, and it would be done as a deed.

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