Nonequilibrium quantum dynamics, such as those in irradiated quantum matter, continue to hold surprises despite decades of significant theoretical and experimental advances. In this talk, I will present our series of results on the nonequilibrium control and engineering of quantum phases using strontium condensates in a strongly driven incommensurate optical lattice. These results include coherently tunable localization phase transitions, interlaced phase diagrams, and Floquet engineering of novel multifractal phases, all controlled by light of variable polarization, even in 1D. We will connect the implications of our findings to the mapping between our 1D system and the 2D irradiated Harper model. Time permitting, I will also discuss our recent experiments in lattice-based atom interferometry and observations of strong-field stabilization.