  Recent developments have produced new ways to encode the data of a torus
equivariant vector bundle over a toric variety using certain representable
matroid(s) labeled by polyhedral data. In this paper we show that this data
makes sense for non-representable matroids as well. We call the resulting
combinatorial objects toric matorid bundles. We define equivariant $K$-theory
and characteristic classes of these bundles. As a particular case, we show that
any matroid comes with tautological toric matroid bundles over the
permutahedral toric variety and the corresponding equivariant K-classes and
Chern classes agree with those in the recent work of Berger-Eur-Spink-Tseng on
matroid invariants. Moreover, in analogy with toric vector bundles, we define
sheaf of sections and Euler characteristic as well as positivity notions such
as global generation, ampleness and nefness for toric matroid bundles. Finally,
we study the splitting of toric matroid bundles and, in particular, an analogue
of Grothendieck's theorem on splitting of vector bundles on the projective
line.
