  Explicit construction of local observable algebras in quasi-Hermitian quantum
theories is derived in both the tensor product model of locality and in models
of free fermions. The latter construction is applied to several cases of a
$\mathcal{PT}$-symmetric toy model of particle-conserving free fermions on a
1-dimensional lattice, with nearest neighbour interactions and open boundary
conditions. Despite the locality of the Hamiltonian, local observables do not
exist in generic collections of sites in the lattice. The collections of sites
which do contain nontrivial observables strongly depends on the complex
potential.
