By "quantum spin system" I mean a physical system with qu-$d$-its (called "spins", for possibly different $d$) distributed somehow over space and a Hamiltonian that is a sum of arbitrary local operators.
When people say that systems "break time-reversal symmetry" or "are chiral" I guess this means that there exists no (anti-)unitary operator acting on the many-body Hilbert space that leaves the ground state/Hamiltonian invariant and that has certain properties (that make it a valid "time-reversal" or "chiral" symmetry operator).
I know there is a definition for free electrons which I guess still works fine with interactions, but I have no idea how to translate this definition to a setting where there isn't even a notion of an occupation basis.
For Levin-Wen string-net models (and other similar fixed-point models), that are said to be "non-chiral", there exists the following symmetry: States can be defined on arbitrary lattices and inherit all orientation-preserving symmetries of the lattice. Reflections of the lattice correspond to complex conjugation. So reflecting the whole lattice and deforming it back via $F$-moves yields a anti-unitary symmetry. Is this symmetry somehow connected to the "chiral" symmetry that the system should have?