  We provide a generalisation of the matrix product operator (MPO) formalism
for string-net projected entangled pair states (PEPS) to include non-unitary
solutions of the pentagon equation. These states provide the explicit lattice
realisation of the Galois conjugated counterparts of (2+1) dimensional TQFTs,
based on tensor fusion categories. Although the parent Hamiltonians of these
renormalisation group fixed point states are non-Hermitian, many of the
topological properties of the states still hold, as a result of the pentagon
equation. We show by example that the the topological sectors of the Yang-Lee
theory (the non-unitary counterpart of the Fibonacci fusion category) can be
constructed even in the absence of closure under Hermitian conjugation of the
basis elements of the Ocneanu tube algebra. We argue that this can be
generalised to the non-unitary solutions of all $SU(2)$ level $k$ models. The
topological sector construction is demonstrated by applying the concept of
strange correlators to the Yang-Lee model, giving rise to a non-unitary version
of the classical hard hexagon model in the Yang-Lee universality class and
obtaining all generalised twisted boundary conditions on a finite cylinder of
the Yang-Lee edge singularity.
