  We revisit anomalous phases related to large gauge transformations, such as
the Witten anomaly. The latter, known to plague $d=4$ $Sp(k)$ theories, is
well-understood in terms of $\pi_4(Sp(k))=\mathbb{Z}_2$, but it also has an
oblique relation to the instantons, labeled by $\pi_3(G)=\mathbb{Z}$, via the
fermion zero mode counting. We revisit this relation and point out how $SU(N)$
theories escape an anomalous sign of the latter type, only thanks to the
perturbative anomaly cancelation condition that restricts the chiral fermion
spectrum. This leads to the question of what happens if the latter, more
mundane anomaly is canceled by an inflow instead. After raising an open
question about fractional D3 probe theories, we explore the simplest bottom-up
model of such a kind, due to Witten and Yonekura, from which we find the
relevant chiral theories to be free of such a disease despite the unrestricted
chiral spectra. We close with a simple but often-overlooked observation about
how fermionic zero modes enter physics differently between Euclidean and
Lorentzian descriptions and point out a related issue in $d=3$.
