There wouldn't be any shelters on unowned land because it would be foolish to improve it if anyone could take your improvements from you at a moment's notice. Now you need a government to enforce exclusive use of the land by a given person, and guess what, that's ownership.
If something can be stolen from you, then you own it. You're trying to have it both ways by playing with semantics.
A land value tax charges taxes for owning land. Now you're saying that it wouldn't be taxes on the land, but rather, taxes on improvements. Then it's no longer a land value tax at all then. The whole point of a land-value tax is that the tax reflects the potential value of the land, regardless of any improvements that might be on it, e.g. the Empire State Building or a flat parking lot would owe the same amount of LVT in midtown Manhattan. What you are proposing is a regular property tax, which does take improvements into account.
I didn't say the shelter was being taxed for value. The shelter is on land. You pay taxes on the land. Ergo, you're paying taxes for the shelter. The only reason you'll probably want to continue paying taxes on the land is because you have a shelter there.
>If something can be stolen from you, then you own it.
Right, you have been grant the right to use the land as long as you pay the taxes, which you may or may not have built a shelter on. You also own the shelter. You don't own the land.
So you're proposing the exact same system that we have now, working in the same way, except that you aren't calling it ownership for some reason, even though being granted the exclusive right to use land by the governemnt, and paying taxes on said land, is what ownership is.
I just don't think that you've thought any of this through, or that you have a coherent image in your mind as to what you're proposing, or how it's different in any way from the status quo.
What's the problem that makes taxing land with no ownership untenable?