It ends with a crash, like always. In 2008 home prices got cut in half. An REIT will overextend, miss loan payments and fail, flooding the market with homes, which will drop the market prices. This will cascade, taking out more REITs.
They'll just get sold and transferred to another large REIT and the house will never hit the market.
The housing market isn't going to crash. I foresee something like Australia happening: the prices just continue to go up, for decades.