There is no geometric morphism $f : E \to F$ if $E$ is a realizaiblity topos and $F$ is a Grothendieck topos. Indeed, if we had such an $f$, then for any indexing set $I$, the topos $E$ would have the $I$-fold coproduct $\coprod_I 1$ of terminal objects: such a coproduct exists in $F$ and $f^*$ will carry it over to $E$, as it preserves the terminal object and coproducts. Taking $I$ to be the underlying pca of $E$ shows that this is impossible (by a cadinality argument, see also the notion of width in Lemma 2.11 in Sheaf toposes for realizability, PDF here).
Regarding the opposite direction, I once heard Peter Johstone state that every geometric morphism from a Grothendieck topos to a realizability topos factors through $(\Gamma \dashv \nabla) : \mathsf{Set} \to \mathsf{RT}(A)$, but I cannot remember why (I was too young to understand much of Peter Johstone anyway).