  We consider 't Hooft anomalies of four-dimensional gauge theories whose
fermion matter content admits $Spin_G(4)$ generalized spin structure, with $G$
either gauged or a global symmetry. We discuss methods to directly compute
$w_2\cup w_3$ 't Hooft anomalies involving Stiefel-Whitney classes of gauge and
flavor symmetry bundles that such theories can have on non-spin manifolds, e.g.
$M_4=\mathbb{CP}^2$. Such anomalies have been discussed for $SU(2)$ gauge
theory with adjoint fermions, where they were shown to give an effect that was
originally found in the Donaldson-Witten topological twist of ${\cal N}=2$ SYM
theory. We directly compute these anomalies for a variety of theories,
including general $G$ gauge theories with adjoint fermions, $SU(2)$ gauge
theory with fermions in general representations, and $Spin(N)$ gauge theories
with fundamental matter. We discuss aspects of matching these and other 't
Hooft anomalies in the IR phase where global symmetries are spontaneously
broken, in particular for general $G_{\rm gauge}$ theory with $N_f$ adjoint
Weyl fermions. For example, in the case of $N_f=2$ we discuss anomaly matching
in the IR phase consisting of $h^\vee _{G_{\rm gauge}}$ copies of a
$\mathbb{CP}^1$ non-linear sigma model, including for the $w_2w_3$ anomalies
when formulated with $Spin_{SU(2)_{\rm global}}(4)$ structure.
