We show that if a memoryless multiple-access channel (MAC) is governed by an independent and identically distributed state sequence, then—unlike the single-user case—the capacity region is typically increased if the state is revealed to the encoders in a strictly causal way. For this scenario, we derive inner and outer bounds on the capacity region. For the Gaussian MAC whose state sequence comprises the channel noise, we compute the capacity region and propose a variation on the Schalkwijk–Kailath scheme that achieves capacity with a double-exponential decay of the maximal probability of error. We also study the causal case for which we derive an achievable region, which is typically strictly larger than the region achievable with naïve Shannon strategies.