  The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a
generalized cone complex that parametrizes metric vertex-weighted graphs of
genus $g$. For each such graph $\Gamma$, the associated canonical linear system
$\vert K_\Gamma\vert$ has the structure of a polyhedral complex. In this
article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and
study its basic combinatorial properties. Our construction is illustrated with
explicit computations and examples.
