Material decomposition in X-ray imaging uses the energy-dependence of attenuation to virtually decompose an object into specific constituent materials. X-ray phase contrast imaging is a developing technique that can enhance image contrast seen from weakly attenuating objects. In this paper, we combine spectral phase contrast imaging with material decomposition to both better visualise weakly attenuating features and separate them from overlying objects in radiography. We derive an algorithm that performs both tasks simultaneously and verify it against numerical simulations and experimental measurements of ideal two-component samples composed of pure aluminium and poly(methyl methacrylate). Additionally, we showcase first imaging results of a rabbit kitten's lung. The attenuation signal of a thorax, in particular, is dominated by the strongly attenuating bones of the ribcage, which combined with the weak soft tissue signal make it difficult to visualise the fine anatomical structures across the whole lung. In all cases, clean material decomposition was achieved, without residual phase contrast effects, from which we generate an un-obstructed image of the lung, free of bones. Spectral propagation-based phase contrast imaging has the potential to be a valuable tool, not only in future lung research, but also in other systems for which phase contrast imaging in combination with material decomposition proves to be advantageous.