Granular beds driven by overlying shear flows begin to erode when the stress delivered to the bed by the fluid exceeds a critical value. Previous studies have shown that this critical stress depends on the stress history of the bed, and that beds will strengthen when subjected to subcritical stresses. By measuring the behavior of erodible beds in a laboratory flume, we confirm this strengthening effect, but also find that it is strongly directional. We find that preconditioned granular beds are indeed more resistant to erosion when driven in the direction of the conditioning flow, but that this strengthening is accompanied by a weakening when driven in other directions. Preconditioned beds are in fact more susceptible to erosion by flows in the direction opposite to that of the conditioning flow than freshly settled beds are. Our results show that the strength of a natural erodible bed with a stress history is likely to be highly anisotropic, with significant implications for predictions of sediment transport.