  We describe the anomaly structure of an composite Higgs model in which the
$SO(5)/SO(4)$ coset structure of the minimal model is extended by an
additional, non-linearly-realized $U(1)_{\eta}$. In addition, we show that the
effective lagrangian admits a term that, like the Wess-Zumino-Witten term in
the chiral lagrangian for QCD, is not invariant under the non-linearly realized
symmetries, but rather changes by a total derivative. This term is unlike the
Wess-Zumino-Witten term in that it does not arise from anomalies. If present,
it may give rise to the rare decay $\eta \rightarrow h W^+ W^- Z$. The
phenomenology of the singlet in this model differs from that in a model based
on $SO(6)/SO(5)$, in that couplings to both gluons and photons, arising via
anomalies, are present. We show that while some tuning is needed to accommodate
flavour and electroweak precision constraints, the model is no worse than the
minimal model in this regard.
