We investigate the existence of a robust, i.e., continuous, representation of the conditional distribution in a stochastic filtering model for multidimensional correlated jump-diffusions. Even in the absence of jumps, it is known that in general such a representation can only be continuous with respect to rough path topologies, leading us naturally to express the conditional dynamics as a rough stochastic differential equation with jumps. Via the analysis of such equations, including exponential moments, Skorokhod continuity, and randomisation of the rough path, we establish several novel robustness results for stochastic filters.