  We consider the $O(3)$ nonlinear sigma model with the $\theta$-term and its
linear counterpart in 1+1D. The model has discrete time-reflection and
space-reflection symmetries at any $\theta$, and enjoys the periodicity in
$\theta\rightarrow \theta+2\pi$. At $\theta=0,\pi$ it also has a
charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection
symmetries is interpreted as putting the theory on the nonorientable $\mathbb
RP^2$ manifold, after which the $2\pi$ periodicity of $\theta$ and the $C$
symmetry at $\theta=\pi$ are lost. We interpret this observation as a mixed 't
Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$
symmetries when $\theta=\pi$. Anomaly matching implies that in this case the
ground state cannot be trivially gapped, as long as $C$, $P$ and $T$ are all
good symmetries of the theory. We make several consistency checks with various
semi-classical regimes, and with the exactly solvable XYZ model. We interpret
this anomaly as an anomaly of the corresponding spin-half chains with
translational symmetry, parity and time reversal (but not involving the
$SO(3)$-spin symmetry), requiring that the ground state is never trivially
gapped, even if $SO(3)$ spin symmetry is explicitly and completely broken. We
also consider generalizations to $\mathbb{C}P^{N-1}$ models and show that the
$C$-$P$-$T$ anomaly exists for even $N$.
