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 $\eta \to hW+W-Z$. 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.