The transverse field Ising model in 1+1d has two phases: a symmetric "disordered" phase and a symmetry-breaking "ordered" phase. Both of these phases have a finite excitation gap. In the symmetry-breaking phase, there is long-range order, that is $\langle Z_i Z_j\rangle \neq 0$ for $|i - j| \rightarrow \infty$. ($Z_i$ is the Pauli $z$ operator on site $i$.)
However, I thought that gapped states should have short-range, i.e., exponentially decaying, correlations. In this answer, Dominic Else says
A ground state of gapped Hamiltonian must have correlations which decay exponentially with distance (this has been proved rigorously by Hastings and Koma).
My question is: why are these facts (long-range correlations in gapped, symmetry-breaking phase and short-range correlations in gapped phase) not in contradiction with each other? I think I must be missing something simple, since both sides of this are well-understood.