Recent experiments demonstrate precise control over coherently excited circular phonon modes using high-intensity terahertz lasers, opening new pathways towards dynamical, ultrafast design of magnetism in functional materials. While the phonon Zeeman effect enables a theoretical description of phonon-induced magnetism, it lacks efficient angular momentum transfer from the phonon to the electron sector. In this work, we put forward a coupling mechanism based on electron-nuclear quantum geometry, with the inverse Faraday effect as a limiting case. This effect is rooted in the phase accumulation of the electronic wave function under a circular evolution of nuclear coordinates. An excitation pulse then induces a transient level splitting between electronic orbitals that carry angular momentum. First-principles simulations on SrTiO3 demonstrate that in parts of the Brillouin zone, this splitting between orbitals carrying angular momentum can easily reach 50 meV.