  Quantum field theory has various projective characteristics which are
captured by what are called anomalies. This paper explores this idea in the
context of fully-extended three-dimensional topological quantum field theories
(TQFTs).
  Given a three-dimensional TQFT (valued in the Morita 3-category of fusion
categories), the anomaly identified herein is an obstruction to gauging a
naturally occurring orthogonal group of symmetries, i.e. we study 't Hooft
anomalies. In other words, the orthogonal group almost acts: There is a lack of
coherence at the top level. This lack of coherence is captured by a "higher
(central) extension" of the orthogonal group, obtained via a modification of
the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension
tautologically acts on the given TQFT/fusion category, and this precisely
classifies a projective (equivalently anomalous) TQFT. We explain the sense in
which this is an analogue of the classical spin representation. This is an
instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is
projective.
  In the appendices we establish a general relationship between the language of
projectivity/anomalies and the language of topological symmetries. We also
identify a universal anomaly associated with any theory which is appropriately
"simple".
