Edge illumination X-ray phase-contrast tomography (EIXPCT) is an imaging technique that estimates the spatially variant X-ray refractive index and absorption distribution within an object while seeking to circumvent the limitations of previous benchtop implementations of X-ray phase-contrast tomography. As with gratingor analyzer-based methods, conventional image reconstruction methods for EIXPCT require that two or more images be acquired at each tomographic view angle. This requirement leads to increased data acquisition times, hindering in vivo applications. To circumvent these limitations, a joint reconstruction (JR) approach is proposed that concurrently produces estimates of the refractive index and absorption distributions from a tomographic data set containing only a single image per tomographic view angle. The JR reconstruction method solves a nonlinear optimization problem by use of a novel iterative gradient-based algorithm. The JR method is demonstrated in both computer-simulated and experimental EIXPCT studies.