  We study the Piatetski-Shapiro construction, which takes a totally real field
F and a Shimura datum (G,X) and produces a new Shimura datum (H,Y). If F is
Galois, then the Galois group Gamma of F acts on (H,Y), and we show that the
Gamma-fixed points of the Shimura varieties for (H,Y) recover the Shimura
varieties for (G,X) under some hypotheses. For Shimura varieties of Hodge type
with parahoric level, we show that the same is true for the p-adic integral
models constructed by Pappas--Rapoport, if p is unramified in F. We also study
the Gamma-fixed points of the Igusa stacks of Zhang for (H,Y) and prove optimal
results.
