At diagnostic X-ray energies, variations in the real component of the refractive index of tissues are several orders of magnitude larger than variations in the imaginary component, or equivalently, the X-ray attenuation coefficient. Consequently, X-ray phase-contrast imaging may permit the visualization of tissues that have identical, or very similar, X-ray absorption properties. Quantitative in-line phase-contrast tomography methods seek to reconstruct the three-dimensional (3D) complex X-ray refractive index distribution of tissue. Almost all existing image reconstruction algorithms for quantitative phase-contrast tomography make physical assumptions that are not consistent with benchtop or clinical implementations that employ an X-ray tube. Such assumptions include a monochromatic plane-wave X-ray beam that possesses perfect coherence properties. In this work, we implement and investigate a reconstruction theory for quantitative phase-contrast tomography that is suitable for use with polychromatic X- ay beams produced by a tube source. An image reconstruction algorithm is implemented that requires, as input data, two intensity measurements at each tomographic view that correspond to incident X-ray beams with distinct coherence properties. Computer-simulation studies that emulate polychromatic tube-based imaging conditions are conducted to assess the effectiveness of the reconstruction method for characterizing soft tissue structures.