We obtain the phase diagrams of field theories of intertwined orders in the presence of periodic driving by an external field which preserves all symmetries. We consider both a conventional Landau theory of competing orders, and a fractionalized theory in which the order parameters are distinct composites of an underlying multi-component Higgs field. We work in the large $N$ limit and couple to a Markovian bath. The long time limits are characterized by non-zero average values, oscillations with the drive period and/or half the drive period, quasi-periodic oscillations, or chaotic behavior.