We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters $q$ and $t$, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials. By specializing $q$ and $t$ in various ways, one recovers both the row and column insertion versions of the Robinson--Schensted correspondence, as well as several $q$- and $t$-deformations of row and column insertion which have been introduced in recent years in connection with integrable probability.