  We study a minimal model involving two species of particles interacting via
quorum-sensing rules. Combining simulations of the microscopic model and linear
stability analysis of the associated coarse-grained field theory, we identify a
mechanism for dynamical pattern formation that does not rely on the standard
route of intra-species effective attractive interactions. Instead, our results
reveal a highly dynamical phase of chasing bands induced only by the combined
effects of self-propulsion and nonreciprocity in the inter-species couplings.
Turning on self-attraction, we find that the system may phase separate into a
macroscopic domain of such chaotic chasing bands coexisting with a dilute gas.
We show that the chaotic dynamics of bands at the interfaces of this
phase-separated phase results in anomalously slow coarsening.
