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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.

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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" for the details.

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