Prismatic $F$-crystals and Wach modules

We show that the category of analytic/completed prismatic $F$-crystals on the absolute prismatic site of a small (unramified at $p$) base ring is naturally equivalent to the category of relative Wach modules from the theory of $(\varphi, \Gamma)$-modules. The result is obtained by showing that the data of the Galois action on a Wach module is equivalent to the data of a prismatic stratification on the underlying $\varphi$-module. Along the way, we obtain new descent results for relative Wach modules.