The type IIB matrix model was proposed as a non-perturbative formulation of superstring theory in 1996. We simulate a model that describes the late time behavior of the IIB matrix model by applying the complex Langevin method to overcome the sign problem. We clarify the relationship between the Euclidean and the Lorentzian versions of the type IIB matrix model in a recently discovered phase. By introducing a constraint, we obtain a model where the spacetime metric is Euclidean at early times, whereas it dynamically becomes Lorentzian at late times.