  Given an (infinite) relational structure $\mathbb S$, we say that a finite
structure $\mathbb C$ is a minimal finite factor of $\mathbb S$ if for every
finite structure $\mathbb A$ there is a homomorphism $\mathbb S\to \mathbb A$
if and only if there is a homomorphism $\mathbb{C} \to \mathbb{A}$. In this
brief note we prove that if CSP($\mathbb S$) is in GMSNP, then $\mathbb S$ has
a minimal finite factor $\mathbb C$, and moreover, CSP($\mathbb C$) reduces in
polynomial time to CSP($\mathbb S$). We discuss two nice applications of this
result. First, we see that if a finite promise constraint satisfaction problem
PCSP($\mathbb A,\mathbb B$) has a tractable GMSNP sandwich, then it has a
tractable finite sandwich. We also show that if $\mathbb G$ is a non-bipartite
(possibly infinite) graph with finite chromatic number, and CSP($\mathbb G$) is
in GMSNP, then CSP($\mathbb G$) in NP-complete, partially answering a question
recently asked by Bodirsky and Guzm\'an-Pro.
