The aim of this work is to investigate the improvement of image quality in few-view grating-based phase-contrast computed tomography (PCCT) applications via compressed sensing (CS) inspired iterative reconstruction on an in vitro mouse model. PCCT measurements are performed on a grating-based PCCT setup using a high-brilliance synchrotron source and a conventional tube source. The sampling density of the data is reduced by a factor of up to 20 and iteratively reconstructed. It is demonstrated that grating-based PCCT intrinsically meets the major conditions for a successful application of CS. Contrast fidelity and the reproduction of details is presented in all reconstructed objects. The feasibility of the iterative reconstruction on data generated with a conventional X-ray source is illustrated on a fluid phantom and a mouse specimen, undersampled by a factor of up to 20.