# $p$-adic lifts of tropical varieties

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.