  A toric vector bundle $\mathcal{E}$ is a torus equivariant vector bundle on a
toric variety. We give a valuation theoretic and tropical point of view on
toric vector bundles. We present three (equivalent) classifications of toric
vector bundles, which should be regarded as repackagings of the Klyachko data
of compatible $\mathbb{Z}$-filtrations of a toric vector bundle: (1) as
piecewise linear maps to space of $\mathbb{Z}$-valued valuations, (2) as
valuations with values in the semifield of piecewise linear functions, and (3)
as points in tropical linear ideals over the semifield of piecewise linear
functions. Moreover, we interpret the known criteria for ampleness and global
generation of $\mathcal{E}$ as convexity conditions on its piecewise linear map
in (1). Finally, using (2) we associate to $\mathcal{E}$ a collection of
polytopes indexed by elements of a certain (representable) matroid encoding the
dimensions of weight spaces of global sections of $\mathcal{E}$. This recovers
and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of
$\mathcal{E}$. This is a follow up paper to arXiv:1806.05613.
