  We study compactifications of subvarieties of algebraic tori defined by
imposing a sufficiently fine polyhedral structure on their non-archimedean
amoebas. These compactifications have many nice properties, for example any k
boundary divisors intersect in codimension k. We consider some examples
including $M_{0,n}\subset\bar M_{0,n}$ (and more generally log canonical models
of complements of hyperplane arrangements) and compact quotients of
Grassmannians by a maximal torus.
