Combined PET/MR imaging allows to incorporate the high-resolution anatomical information delivered by MRI into the PET reconstruction algorithm for improvement of PET accuracy beyond standard corrections. We used the working hypothesis that glucose uptake in adipose tissue is low. Thus, our aim was to shift <sup>18</sup>F-FDG PET signal into image regions with a low fat content. Dixon MR imaging can be used to generate fat-only images via the water/fat chemical shift difference. On the other hand, the Origin Ensemble (OE) algorithm, a novel Markov chain Monte Carlo method, allows to reconstruct PET data without the use of forward- and back projection operations. By adequate modifications to the Markov chain transition kernel, it is possible to include anatomical <i>a priori </i>knowledge into the OE algorithm. In this work, we used the OE algorithm to reconstruct PET data of a modified IEC/NEMA Body Phantom simulating body water/fat composition. Reconstruction was performed 1) natively, 2) informed with the Dixon MR fat image to down-weight <sup>18</sup>F-FDG signal in fatty tissue compartments in favor of adjacent regions, and 3) informed with the fat image to up-weight <sup>18</sup>F-FDG signal in fatty tissue compartments, for control purposes. Image intensity profiles confirmed the visibly improved contrast and reduced partial volume effect at water/fat interfaces. We observed a 17±2% increased SNR of hot lesions surrounded by fat, while image quality was almost completely retained in fat-free image regions. An additional <i>in vivo </i>experiment proved the applicability of the presented technique in practice, and again verified the beneficial impact of fat-constrained OE reconstruction on PET image quality.