In this paper, we study spin systems, probabilistic cellular automata and interacting particle systems, which are Markov processes with state space $\{0, 1\}^{\mathbf{Z}^n}$. Restricting ourselves to attractive systems, we consider the stationary processes obtained when either of two distinguished stationary distributions is used, the smallest and largest stationary distributions with respect to a natural partial order on measures. In discrete time, we show that these stationary processes with state space $\{0, 1\}^{\mathbf{Z}^n}$ and index set $\mathbf{Z}$ are isomorphic (in the sense of ergodic theory) to an independent process indexed by $\mathbf{Z}$. In the translation invariant case, we prove the stronger fact that these stationary processes, viewed as $\{0, 1\}$-valued processes with index set $\mathbf{Z}^n \times \mathbf{Z}$ (space-time), are isomorphic to an independent process also indexed by $\mathbf{Z}^n \times \mathbf{Z}$. Such processes are called Bernoulli shifts. Finally, we extend all of these results to continuous time.