What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}\_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions to its deformation to a (toric or log-smooth) variety over a $p$-adic field classified by? In which major cases are they known to vanish? What are the corresponding deformation spaces?


Summary and references would be really appreciated.



$p$-adic lifts of tropical varieties
There are moduli spaces of tropical lifts. They satisfy "Murphy's law": Any behavior which can happen on a scheme of finite type can happen on them. See Katz and Payne "[Realization spaces for tropical fans](https://arxiv.org/abs/0909.4582)" for the details.


