Solid-state batteries are poised to revolutionize the energy landscape, offering higher energy density and improved safety compared to current liquid electrolyte batteries. However, interfacial stability challenges must still be overcome before the most promising materials can become widely accessible. The overwhelming consensus in the literature indicates that existing theoretical frameworks fail to adequately capture multiphysics phenomena, particularly in relation to electrochemical-mechanical couplings. Moreover, magnetic couplings are almost universally neglected despite groundbreaking experimental evidence demonstrating that properly oriented magnetic fields can significantly enhance interfacial stability. To address these gaps, this work proposes a novel, thermodynamically consistent, Lagrangian continuum theory to analytically describe the behavior of battery materials and their interfaces under the coupled influence of thermal, electrochemical, magnetic, and mechanical fields. The resulting constitutive models align well with established theoretical approaches from other contexts, yet allow for additional couplings rarely or never considered in the battery literature, particularly the influence of evolving magnetic fields on ion transport via the Hall and Kelvin force effects. Critically, the theory is computationally implemented using the finite element method. Elements are formulated with curl-conforming edge degrees of freedom to interpolate the magnetic potential and standard Lagrangian interpolation for other fields. Additionally, a block preconditioning methodology is adopted to enable robust, monolithic iterative solutions that effectively capture the cross-influence of magnetic fields on electrochemical-mechanical processes within a battery. As an illustrative example, a charge cycle is simulated for a battery stack with a dendritic imperfection on the anode surface. By observing how the application of a magnetic field alters the current distribution and Lorentz forces near a dendrite tip, the magnetic field necessary to suppress dendrite growth can be estimated. More generally, the proposed technology enables rigorous, analytical exploration of broad classes of materials under very general conditions.