Backfiring Bosonisation

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.