A common rheological hypothesis, that the stress in a fluid element is only a function of its own deformation history, is rendered questionable in liquid‐crystalline polymers (LCPs) due to the presence of distortional elasticity, through which neighboring fluid elements may directly influence one another. However, the fine defect texture in LCPs has led to the suggestion that fluid properties may be averaged over a mesoscopic length scale, intermediate between the molecular and macroscopic, so that averaged measures of fluid structure and stress at this scale depend only on their own deformation history [R. G. Larson and M. Doi, J. Rheol. 35, 539 (1991)]. We describe an experimental test of this hypothesis. If true, it should be possible to use rheological and rheo‐optical data obtained in simple shear flow to predict the velocity and molecular orientation fields in a nonhomogeneous shear flow. Quantitative flow birefringence experiments are conducted on a liquid‐crystalline solution of poly(benzyl glutamate), in plane Poiseuille flow. At low flow rates, the data support the local response hypothesis. As flow rate is increased, however, a profound instability occurs that is unanticipated based on behavior reported in homogeneous simple shear flow. The instability is characterized by large wavelength disturbances in structure oriented perpendicular to the flow direction that are clearly visible to the naked eye. With increasing flow rate, these structures decrease in size and become increasingly chaotic. Despite the onset of the instability, time‐averaged measurements of average orientation may be qualitatively predicted based on simple shear flow data.