The Wess-Zumino consistency condition allows more exotic forms of anomalies
than those we usually encounter. For example in two-dimensional conformal field
theories in the curved background with space-time dependent coupling constant
$\lambda^i(x)$, a $U(1)$ current could possess anomalous divergence of the form
$D^\mu J_\mu = \tilde{c} R + \chi_{ij} \partial^\mu \lambda^i
\partial_\mu\lambda_j + \tilde{\chi}_{ij} \epsilon^{\mu\nu} \partial_\mu
\lambda^i \partial_\nu \lambda^j + \cdots $. Another example is the CP odd
Pontryagin density in four-dimensional Weyl anomaly. We could, however, argue
that they are impossible in conformal field theories because they cannot
correspond to any (unregularized) conformally invariant correlation functions.
We find that this no-go argument may be a red herring. We show that some of
these impossible anomalies avoid the no-go argument because they are not
primary operators, and the others circumvent it because they are realized as
semi-local terms as is the case with the conformally invariant Green-Schwartz
mechanism and in the higher dimensional analogue of Liouville or linear dilaton
theory.