  We associate a matroid $M(\widetilde{\Gamma}/\Gamma)$ to a harmonic double
cover $\pi:\widetilde{\Gamma}\to \Gamma$ of metric graphs. The matroid
$M(\widetilde{\Gamma}/\Gamma)$ is a geometric interpretation of Zaslavsky's
signed graphic matroid. We show that the principalization
$\mathrm{Prym}_p(\widetilde{\Gamma}/\Gamma)$ of the tropical Prym variety of
the double cover can be reconstructed from $M(\widetilde{\Gamma}/\Gamma)$,
equipped with certain additional decorations. We describe the simplification of
the matroid $M(\widetilde{\Gamma}/\Gamma)$ and show that the Prym variety does
not change under simplification.
