Sale-leaseback
You sell the house, get the equity, and rent it back. Title leaves.
Two contracts
The sale is a sale. Net proceeds equal price minus mortgage payoff minus costs, the same math as listing the house. A separate lease lets you stay, commonly 6 to 36 months, sometimes with a repurchase option at a set formula. Rent replaces the mortgage payment.
You no longer own the appreciation. That is the cost of taking all of the equity without moving the same day.
Educational only. Not a lender, and not financial, tax, or legal advice.