Summary: The set of orbits of \(\text{GL}(V)\) in \(\text{Fl}(V)\times\text{Fl}(V)\times V\) is finite, and is parametrized by the set of certain decorated permutations in a work of P. Magyar, J. Weyman and A. Zelevinsky [Adv. Math. 141, No. 1, 97-118 (1999; Zbl 0951.14034)]. We describe a mirabolic Robinson-Shensted-Knuth correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of \(\text{GL}(V)\) arising from \(\text{Fl}(V)\times\text{Fl}(V)\times V\). We also give conjectural applications to the classification of unipotent mirabolic character sheaves on \(\text{GL}(V)\times V\).