  It is known that some theories of class $S$ are actually factorized into
multiple decoupled nontrivial four-dimensional $N=2$ theories. We propose a way
of constructing examples of this phenomenon using the physics of half-BPS
surface defects, and check that it works in one simple example: it correctly
reproduces a known realization of two copies of $N=2$ superconformal $SU(2)$
QCD, describing this factorized theory as a class $S$ theory of type $A_3$ on a
five-punctured sphere with a twist line. Separately, we also present explicit
checks that the Coulomb branch of a putative factorized class $S$ theory has
the expected product structure, in two examples.
