Phase and polarization of coherent light are highly perturbed by interaction with microstructural changes in premalignant tissue, holding promise for label-free detection of early tumors in endoscopically accessible tissues such as the gastrointestinal tract. Flexible optical multicore fiber (MCF) bundles used in conventional diagnostic endoscopy and endomicroscopy scramble phase and polarization, restricting clinicians instead to low-contrast amplitude-only imaging. We apply a transmission matrix characterization approach to produce full-field en-face images of amplitude, quantitative phase, and resolved polarimetric properties through an MCF. We first demonstrate imaging and quantification of biologically relevant amounts of optical scattering and birefringence in tissue-mimicking phantoms. We present an entropy metric that enables imaging of phase heterogeneity, indicative of disordered tissue microstructure associated with early tumors. Finally, we demonstrate that the spatial distribution of phase and polarization information enables label-free visualization of early tumors in esophageal mouse tissues, which are not identifiable using conventional amplitude-only information.