We propose a mechanism for inducing a Turing instability in systems whose only stable state is pattern-free and homogeneous. Global alternation between two dynamics, each of which has the same homogeneous stable state, may induce a Turing instability that leads to pattern formation. We determine what kind of alternation can drive the system to a Turing instability, and show that the appearance of the induced spatiotemporal structure depends on the ratio of two characteristic times, one determined by the external forcing and the other by the instability that drives the system at short times. The mechanism is illustrated by means of theoretical calculations and numerical simulations on two well-known biological models that are relevant in morphogenesis.