What is the ruling on purchasing a house through a fixed annual increase on the principal amount, owned by a non-Muslim party, and registered in the wife's name?

If your husband's relative buys the house for herself and then sells it to you in installments after taking possession of it, there is no objection to her increasing the price. The owner may sell what he owns for a profit, as Urwah al-Bariqi did when he sold one of the two sheep for a dinar. Her promise to waive the five years does not obligate her to fulfill it, according to the majority of scholars, and it is permissible for you to accept it. There is no objection to registering the house in your wife's name in the administrative documentation as long as you agree, and it is preferable to have witnesses to your ownership of it.