  We prove that endowing a real-time probabilistic or quantum computer with the
ability of postselection increases its computational power. For this purpose,
we provide a new model of finite automata with postselection, and compare it
with the model of L\={a}ce et al. We examine the related language classes, and
also establish separations between the classical and quantum versions, and
between the zero-error vs. bounded-error modes of recognition in this model.
