"Alas, that would be a terribly idea, because all of a sudden you have the government snooping into your living arrangements. Especially for grey areas where people share their time between multiple apartments. Way too complicated."

... not really. Most folks only own one house, so nothing complicated. A few folks own multiple houses. These folks already need to establish residency in one house for tax purposes in a lot of places: Taxes vary from city to city and state to state. It isn't a new grey area, and we can use this established system to tax. The folks can, in general, just use one house or apartment or pay more for more than one house.