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The dynamics and the structure of the solar corona are determined by its magnetic field. Measuring coronal magnetic fields is, however, extremely hard. The polarization of low-frequency radio emissions has long been recognized as one of the few effective observational probes of magnetic fields in the mid and high corona. However, the extreme intrinsic variability of this emission, the limited ability of most of the available existing instrumentation (until recently) to capture it, and the technical challenges involved have all contributed to its use being severely limited. The high dynamic-range spectropolarimetric snapshot imaging capability that is needed for radio coronal magnetography is now within reach. This has been enabled by the confluence of data from the Murchison Widefield Array (MWA), a Square Kilometre Array (SKA) precursor, and our unsupervised and robust polarization calibration and imaging software pipeline dedicated to the Sun-Polarimetry using the Automated Imaging Routine for Compact Arrays of the Radio Sun (P-AIRCARS). Here, we present the architecture and implementation details of P-AIRCARS. Although the present implementation of P-AIRCARS is tuned to the MWA, the algorithm itself can easily be adapted for future arrays, such as SKA1-Low. We hope and expect that P-AIRCARS will enable exciting new science with instruments like the MWA, and that it will encourage the wider use of radio imaging in the larger solar physics community.