Polarized light microscopy is a widely used technique to observe specimens that are optically anisotropic, or birefringent. It has a broad applicability in the study of minerals, soft materials such as polymers, complex fluids or liquid crystals, and organic tissues in biology and medicine. Most of these observations are qualitative in nature, as it is not obvious to quantify the spatial distribution of optical anisotropy of specimens. Moreover, existing commercial implementations for quantitative polarimetry are costly and slow in nature, precluding real time observation of dynamical processes. Here, we present a custom-made implementation of an optical microscope for quantitative polarimetry at the cost of a standard scientific polarizing microscope. The instrument allows to extract the local optical axis and birefringence of transparent materials with a frequency of several Hz. The instrument is built using off-the-shelf optomechanical components, which optimizes cost, availability, and modularity. An example of the latter is the fact that we combine the polarimetry measurements with simultaneous fluorescence microscopy, which results in a powerful multimodal instrument with broad potential applications.