  Non-equilibrium driving systems provide fertile ground for exploring
intriguing spontaneous symmetry breaking phenomena. In this study, we report on
the intertwined discrete spacetime translational symmetry breaking in a
driven-dissipative spin system without pure spatial translational symmetry,
resulting in the emergence of an out-of-equilibrium nonmagnetic phase termed
the "discrete spacetime crystal." In contrast to the previously widely studied
discrete time crystal, the spins establish a distinctive long-range crystalline
order not only in time, but also in the intertwined spacetime direction. We
further demonstrate the exponentially-long lifetime of the spatial-temporal
order against generic weak environmental noise fluctuations. Finally, possible
experimental realizations of our model are discussed. Our findings shed light
on the existence of exotic phases of matter characterized by intertwined
space-time symmetry breaking, paving the way for future investigations into the
properties and potential applications of these exotic states.
