Exoplanets are often associated with disks of dust and debris, analogs of the Kuiper Belt in our solar system. These "debris disks" show a variety of non-trivial structures attributed to planetary perturbations and utilized to constrain the properties of the planets. However, analyses of these systems have largely ignored the fact that, increasingly, debris disks are found to contain small quantities of gas, a component all debris disks should contain at some level. Several debris disks have been measured with a dust-to-gas ratio around unity where the effect of hydrodynamics on the structure of the disk cannot be ignored. Here we report that dust-gas interactions can produce some of the key patterns seen in debris disks that were previously attributed to planets. Through linear and nonlinear modeling of the hydrodynamical problem, we find that a robust clumping instability exists in this configuration, organizing the dust into narrow, eccentric rings, similar to the Fomalhaut debris disk. The hypothesis that these disks might contain planets, though thrilling, is not necessarily required to explain these systems.