The long chain of a polymer molecule often has intrinsic linear birefringence and/or dichroism, which lead to macroscopic properties when the molecules have a preferred orientation under nonequilibrium conditions, such as flow of a polymer melt. Measurements of properties, including linear birefringence, in a polymer melt can in turn lead to valuable insights into its structure, morphology, and rheology. We demonstrate that a spectropolarimeter based on an acousto-optic tunable filter can measure linear birefringence in real time during the manufacture of polymers, at hundreds of wavelengths, within 5 min, and thus provides more information than use of a spectrometer or laser-based polarimetry alone. In this work, optical properties of different molten ethylene vinyl acetate samples were measured with a spectropolarimeter based on an acousto-optic tunable filter and a photoelastic modulator in the near infrared. Regression models of the results indicate good correlation between the melt indices of the samples and the measured spectra.