  Let $X$ be a variety with a stratification $\mathcal{S}$ into smooth locally
closed subvarieties such that $X$ is locally a product along each stratum
(e.g., a symplectic singularity). We prove that assigning to each open subset
$U \subset X$ the set of isomorphism classes of locally projective crepant
resolutions of $U$ defines an $\mathcal{S}$-constructible sheaf of sets. Thus,
for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the
set of germs of projective crepant resolutions at $s$, leaving invariant the
germs extending to the entire stratum. Global locally projective crepant
resolutions correspond to compatible such choices for all strata. For example,
if the local projective crepant resolutions are unique, they automatically glue
uniquely.
  We give criteria for a locally projective crepant resolution $\rho: \tilde X
\to X$ to be globally projective. We show that the sheafification of the
presheaf $U \mapsto \text{Pic}(\rho^{-1}(U)/U)$ of relative Picard classes is
also constructible. The resolution is globally projective only if there exist
local relatively ample bundles whose classes glue to a global section of this
sheaf. The obstruction to lifting this section to a global ample line bundle is
encoded by a gerbe on the singularity $X$. We show the gerbes are automatically
trivial if $X$ is a symplectic quotient singularity. Our main results hold in
the more general setting of partial crepant resolutions, that need not have
smooth source. We apply the theory to symmetric powers and Hilbert schemes of
surfaces with du Val singularities, finite quotients of tori, multiplicative
and Nakajima quiver varieties, as well as to canonical threefold singularities.
