'Debris disks' around young stars (analogues of the Kuiper Belt in our Solar System) show a variety of non-trivial structures attributed to planetary perturbations and used to constrain the properties of those planets(1-3). However, these analyses have largely ignored the fact that some debris disks are found to contain small quantities of gas(4-9), a component that all such disks should contain at some level(10,11). Several debris disks have been measured with a dust-to-gas ratio of about unity(4-9), at which the effect of hydrodynamics on the structure of the disk cannot be ignored(12,13). Here we report linear and nonlinear modelling that shows that dust-gas interactions can produce some of the key patterns attributed to planets. We find a robust clumping instability that organizes the dust into narrow, eccentric rings, similar to the Fomalhaut debris disk(14). The conclusion that such disks might contain planets is not necessarily required to explain these systems.