  This paper initiates a research program that seeks to recover
algebro-geometric Galois representations from combinatorial data. We study
tropicalizations equipped with symmetries coming from the Galois-action present
on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric
varieties. Over a Henselian field, the resulting tropicalization maps become
Galois-equivariant. We call their images Galois-equivariant tropicalizations,
and use them to construct a large supply of Galois representations in the
tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and
Zharkov. We also prove two results which say that under minimal hypotheses on a
variety $X_{0}$ over a Henselian field $K_{0}$, Galois-equivariant
tropicalizations carry all of the arithmetic structure of $X_{0}$. Namely: (1)
The Galois-orbit of any point of $X_{0}$ valued in the separable closure of
$K_{0}$ is reproduced faithfully as a Galois-set inside some Galois-equivariant
tropicalization of our variety. (2) The Berkovich analytification of $X_{0}$
over the separable closure of $K_{0}$, equipped with its canonical
Galois-action, is the inverse limit of all Galois-equivariant tropicalizations
of our variety.
