We present new recursion operators for (shadows of nonlocal) symmetries of the 4D Martínez Alonso-Shabat equation uty = uzuxy - uyuxz, and we show that their actions can produce new symmetries which are not contained in the Lie algebra of nonlocal symmetries presented in Krasil'shchik and Vojčák (2021). To this end, we construct a non-Abelian covering of the equation in question using the Lax pair with two non-removable parameters.