Tropical geometry allows varieties, and their compatifications and degenerations, to be studied using combinatorial and polyhedral techniques. While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles. I will discuss joint work with Felipe Rincon that gives a definition for of a subscheme of a tropical toric variety. This builds on work of Jeff and Noah Giansiracusa on tropicalizing subschemes, and uses the theory of valuated matroids.