Let us consider the surface of $(3+1)$-dimensional topological insulator, which is protected by the charge conservation $U(1)_Q$ and a time-reversal symmetry $\mathbb{Z}_2^T$. Such a surface, if not breaking $U(1)_Q\rtimes\mathbb{Z}_2^T$ explicitly, cannot be gapped with a unique ground state. Such an ``ingappability'' can be understood by either the nontrivial $U(1)_Q\rtimes\mathbb{Z}_2^T$ symmetry-protected-topological (SPT) bulk or the $U(1)_Q\rtimes\mathbb{Z}_2^T$ symmetry anomaly of itself.
If we explicitly break $\mathbb{Z}_2^T$ only (namely without breaking $U(1)_Q$) on the surface by $(2+1)$-dimensional local interaction to gap it with a unique ground state, such a $\mathbb{Z}_2^T$-explicitly-broken gapped surface state will develop a half Hall conductance.
However, purely $(2+1)$-dimensional electronic system with a unique ground state must have integer Hall conductance. In this sense, this $\mathbb{Z}_2^T$-explicitly-broken surface still needs to be attached to a bulk. My question is, does it mean that this $\mathbb{Z}_2^T$-explicitly-broken gapped surface is still "anomalous" since it cannot be consistent (e.g. non-integer Hall conductance) without a higher dimensional bulk? If so, does it contradict to the fact that $(2+1)$-dimensional Dirac fermion is actually anomaly-free if we only impose the $U(1)_Q$ symmetry without $\mathbb{Z}_2^T$? It seems such a gapped phase is a new phase since it cannot be consistent without a $U(1)_Q$-trivial SPT bulk and this bulk is a nontrivial $U(1)_Q\rtimes\mathbb{Z}_2^T$ bulk if further $\mathbb{Z}_2^T$ is imposed.