  We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a
smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely
valued field with perfect residue field. Our main result gives an equivalence
between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$
and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local
systems on $\mathfrak{X}_\eta$: those of prismatically good reduction.
Additionally, we develop a shtuka realization functor for (analytic) prismatic
$F$-crystals on $p$-adic (formal) schemes and show it satisfies several
compatibilities with previous work on the Tannakian theory of shtukas over such
objects.
