The INS grants citizenship to foreigners on the queue who have a clean criminal record, and for whom a US employer agrees to buy a house (for at least $150,000) as an advance on their salaries.
To make US employers willing to do this, presumably these would be moderate to highly skilled immigrants. But I bet there are at least 400,000 potential immigrants in the world who qualify. So this alone would knock out about 10% of the existing housing inventory.
BTW if any reader who is also a writer finds this intriguing, feel free to run with it. I won't have time to develop this further.
One final thought: You might worry and say, "Well we'd have to require also that the official owner--whether the employer or the immigrant--doesn't try to resell the house for x years."
I actually don't think that's correct, since it can't hurt to allow people voluntary trades. I.e. once the immigrant is in the country and has a house, from that point on if you allow him to sell to a willing American buyer, then that buyer's gain offsets whatever harm accrues to other American house sellers.
Really the solution works by bringing in more people who need housing, whether purchased or rented. But by directly tying the immigration to a house purchase, I think it would be a lot more popular politically.