We propose an elementary tropical analogue of a reductive group that combines the datum of a Weyl group and the tropicalization of a fixed maximal torus. For the classical groups, as well as $G_2$, these tropical reductive groups admit descriptions as tropical matrix groups that resemble their classical counterparts. Employing this perspective, we introduce tropical principal bundles on metric graphs and study their explicit presentations as pushforwards of line bundles along covers with symmetries and extra data. Our main result identifies the essential skeleton of the moduli space of semistable principal bundles on a Tate curve with its tropical analogue.