  Irradiation with light provides a powerful tool to interrogate, control or
induce new quantum states of matter out of equilibrium, however a microscopic
understanding of light-matter coupling in interacting electron systems remains
a profound challenge. Here, we show that light grants a new quantum-geometric
handle to steer and probe correlated quantum materials, whereby photons can
couple directly to the shape and center of the maximally-localized Wannier
functions that comprise the material's interacting bands, dressing both
electronic motion and electronic interactions with light. Notably, this effect
is generic to any material and purely geometric in origin, but dominates
emergent optical responses in correlated electron systems with poorly localized
or obstructed Wannier functions. Spectroscopic consequences are first
illustrated for a paradigmatic strongly interacting model with a tunable
Wannier obstruction. We then present ramifications for non-equilibrium control
of moir\'e heterostructures and find that subjecting magic-angle twisted
bilayer graphene to weak THz radiation can conspire with a fragile topological
obstruction to profoundly alter the material's competing interactions and tune
across boundaries to competing phases.
