Let $P\subset\Bbb R^d$ be a convex polytope. Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations. I am interested in polytopes for which this space is compact. For example, this is the case for polytopes that have a single (or a finite number) of realizations up to projective transformations. But are there any others?
Question: Are there convex polytopes for which the realization space modulo projective transformations is compact but not discrete?
I am aware of the universality theorem for convex polytopes which states that I can choose an arbitrary semi-algebraic set $S$ and find a polytope whose realization space is $S$ up to stable equivalence. But I believe that stable equivalence does not preserves compactness, so this is of limited help.
My question is partially motivated from the observation that polytopes in dimension $d\le 3$ have a reduced realization space that is either discrete (in fact, a single point) or non-compact. The reason is that if such a polytope $P$ is not projectively unique, then one can choose an edge $e\subset P$ and a sequence of realizations where the edge $e$ becomes shorter and shorter, while all other edges stay of length bounded away from zero; so in the limit the polytope will change combinatorial type. This is obvious in 2D, but needs some work in 3D (based on specifically 3-dimensional tools that do not generalize to $d\ge 4$).
