Money. The shares you are talking about already exist, and they're called "money". Of course, the relative value of shares wrt. the house will fluctuate over time and you'd have to account for this when transferring between houses, except this is calculated for you by the market when you just use money.

That sounds like it would be hideously complicated in practice.

Renting works fine if you have some basic regulations (which admittedly is the opposite of the case in the US).