Bioresorbable Mg-based alloys with low density, low elastic modulus, and excellent biocompatibility are outstanding candidates for temporary orthopedic implants. Coincidentally, metal additive manufacturing (AM) is disrupting the biomedical sector by providing fast access to patient-customized implants. Due to the high cooling rates associated with fusion-based AM techniques, they are often described as rapid solidification processes. However, conclusive observations or rapid solidification in metal AM — attested by drastic microstructural changes induced by solute trapping, kinetic undercooling, or morphological transitions of the solid-liquid interface — are scarce. Here we study the formation of banded microstructures during laser powder-bed fusion (LPBF) of a biomedical-grade Magnesium-rare earth alloy, combining advanced characterization and state-of-the-art thermal and phase-field modeling. Our experiments unambiguously identify microstructures as the result of an oscillatory banding instability known from other rapid solidification processes. Our simulations confirm that LPBF-relevant solidification conditions strongly promote the development of banded microstructures in a Mg-Nd alloy. Simulations also allow us to peer into the sub-micrometer nanosecond-scale details of the solid-liquid interface evolution giving rise to the distinctive banded patterns. Since rapidly solidified Mg alloys may exhibit significantly different mechanical and corrosion response compared to their cast counterparts, the ability to predict the emergence of rapid solidification microstructures (and to correlate them with local solidification conditions) may open new pathways for the design of bioresorbable orthopedic implants, not only fitted geometrically to each patient, but also optimized with locally-tuned mechanical and corrosion properties.