  For a fermionic quantum field theory in $d=1+1$ dimensions, there is a subtle
difference between summing over spin structures and gauging $(-1)^F$. If the
gravitational anomaly vanishes mod 16, then both operations are equivalent and
yield a bosonic theory. But if the gravitational anomaly only vanishes mod 8,
then only gauging $(-1)^F$ is allowed, and the result is a fermionic theory.
Our goal is to understand in detail how this happens, despite the fact $(-1)^F$
is defined in terms of shifting the spin structure, which would na\"ively
suggest that both operations are equivalent. We do this from three
perspectives: an abstract view in terms of anomalies, explicit CFT
calculations, and a Symmetry TFT perspective. To conclude, we illustrate our
results using the heterotic string and the famous self-triality of 8
Majorana-Weyl fermions.
