Sand dunes, which arise spontaneously due to the dynamical interplay between a sedimentary interface and a fluid flow, are one of the most famous examples of emergence in a geological system. The large scale organization of a dune field is believed to be controlled by pairwise (either remote or direct) dune-dune interactions. Recent studies have shown that remote long-range feedback is closely related to the turbulent wake structure forming downstream of a dune. Here, we study the stability of an idealized two-dune system arising as a consequence of such remote, wake-induced interactions. The system is realized in a subaqueous quasi-2D laboratory experiment and the results are compared with a qualitative dynamical systems model. Despite its simplicity, the system exhibits rich dynamical behavior. In particular, we show that, depending on the parameter regime, the dune-dune feedback can either stabilize or destabilize the symmetric dune configuration, and we demonstrate the existence of an asymmetric attracting state coupling dunes of different sizes.