Recent advances in structured illumination are enabling a wide range of applications from imaging to metrology, which can benefit from advanced beam characterization techniques. Solving uniquely for the spatial distribution of polarization in a beam typically involves the use of two or more polarization optics, such as a polarizer and a waveplate, which is prohibitive for some wavelengths outside of the visible spectrum. We demonstrate a technique that circumvents the use of a waveplate by exploiting extended Gerchberg-Saxton phase retrieval to extract the phase. The technique enables high-resolution, wavefront-sensing, full-field polarimetry capable of solving for both simple and exotic polarization states, and moreover, is extensible to shorter wavelength light.