Flexible optical fibres, used in conventional medical endoscopy and industrial inspection, scramble phase and polarisation information, restricting users to amplitude-only imaging. Here, we exploit the near-diagonality of the multi-core fibre (MCF) transmission matrix in a parallelised fibre characterisation architecture, enabling accurate imaging of quantitative phase (error <0.3 rad) and polarisation-resolved (errors <10%) properties. We first demonstrate accurate recovery of optical amplitude and phase in two polarisations through the MCF by measuring and inverting the transmission matrix, and then present a robust Bayesian inference approach to resolving 5 polarimetric properties of samples. Our method produces high-resolution (9.0±2.6µm amplitude, phase; 36.0±10.4µm polarimetric) full-field images at working distances up to 1mm over a field-of-view up to 750×750µm2 using an MCF with potential for flexible operation. We demonstrate the potential of using quantitative phase for computational image focusing and polarisation-resolved properties in imaging birefringence.