Synthetic medical data offers a scalable solution for training robust models, but significant domain gaps limit its generalizability to real-world clinical settings. This paper addresses the challenge of cross-domain translation between synthetic and real X-ray images of the head, focusing on bridging discrepancies in attenuation behavior, noise characteristics, and soft tissue representation. We propose MedShift, a unified class-conditional generative model based on Flow Matching and Schrodinger Bridges, which enables high-fidelity, unpaired image translation across multiple domains. Unlike prior approaches that require domain-specific training or rely on paired data, MedShift learns a shared domain-agnostic latent space and supports seamless translation between any pair of domains seen during training. We introduce X-DigiSkull, a new dataset comprising aligned synthetic and real skull X-rays under varying radiation doses, to benchmark domain translation models. Experimental results demonstrate that, despite its smaller model size compared to diffusion-based approaches, MedShift offers strong performance and remains flexible at inference time, as it can be tuned to prioritize either perceptual fidelity or structural consistency, making it a scalable and generalizable solution for domain adaptation in medical imaging. The code and dataset are available at https://caetas.github.io/medshift.html