  Time-dependent driving holds the promise of realizing dynamical phenomenon
absent in static systems. Here, we introduce a correlated random driving
protocol to realize a spatiotemporal order that cannot be achieved even by
periodic driving, thereby extending the discussion of time translation symmetry
breaking to randomly driven systems. We find a combination of temporally
disordered micro-motion with prethermal stroboscopic spatiotemporal long-range
order. This spatiotemporal order remains robust against generic perturbations,
with an algebraically long prethermal lifetime where the scaling exponent
strongly depends on the symmetry of the perturbation, which we account for
analytically.
