Limited-angle computed tomography (CT) is common in industrial applications, where incomplete projection data can cause artifacts. For objects made from homogeneous materials, we propose a joint reconstruction and segmentation method that performs joint image reconstruction and segmentation directly on the projection data. We describe an alternating minimization algorithm to solve the resulting optimization problem, and we modify the primal-dual hybrid gradient algorithm for the non-convex piecewise constant Mumford-Shah model, which is a popular approximation model in biomedical image segmentation. The effectiveness of the proposed approach is validated by simulation and by application to actual micro-CT data sets.