The shear rheology of a binary polymer blend exhibiting a lower critical solution temperature (LCST) phase diagram and a small dynamic asymmetry (difference of glass transition temperatures between its constituents) has been investigated in the vicinity of phase separation; it is a mixture of a random copolymer of styrene and maleic anhydrite and poly(methyl methacrylate). In the linear viscoelastic regime, the material functions are sensitive to phase separation, and the effects of critical concentration fluctuations, which dominate the mechanical response, are quantified, yielding both the binodal and spinodal curves. The weak dynamic asymmetry is apparently responsible for the reduced magnitude of the observed effects, compared to blends exhibiting much larger contrast in glass transition; therefore, this property affects to some degree the accuracy of the theologically determined phase diagram. The steady shear properties are weakly sensitive to phase separation, and suggest that shear-induced demixing may be possible. They also indicate the importance of the amount of strain energy introduced to the blend in controlling the effects of flow on phase behavior. This investigation demonstrates that the universal effects of concentration fluctuations can be detected in LCST binary polymer blends, provided that some dynamic asymmetry exists, and further they can be quantified in order to characterize the interplay between rheology and thermodynamics of these systems.