  The first steps in defining tropicalization for spherical varieties have been
taken in the last few years. There are two parts to this theory: tropicalizing
subvarieties of homogeneous spaces and tropicalizing their closures in
spherical embeddings. In this paper, we obtain a new description of spherical
tropicalization that is equivalent to the other theories. This works by
embedding in a toric variety, tropicalizing there, and then applying a
particular piecewise projection map. We use this theory to prove that taking
closures commutes with the spherical tropicalization operation.
