  Topological characterization of non-Hermitian band structures demands more
than a straightforward generalization of the Hermitian cases. Even for
one-dimensional tight-binding models with nonreciprocal hopping, the appearance
of point gaps and the skin effect leads to the breakdown of the usual
bulk-boundary correspondence. Luckily, the correspondence can be resurrected by
introducing a winding number for the generalized Brillouin zone for systems
with even number of bands and chiral symmetry. Here, we analyze the topological
phases of a nonreciprocal hopping model on the stub lattice, where one of the
three bands remains flat. Due to the lack of chiral symmetry, the biorthogonal
Zak phase is no longer quantized, invalidating the winding number as a
topological index. Instead, we show that a $Z_2$ invariant can be defined from
Majorana's stellar representation of the eigenstates on the Bloch sphere. The
parity of the total azimuthal winding of the entire Majorana constellation
correctly predicts the appearance of edge states between the bulk gaps. We
further show that the system is not a square-root topological insulator,
despite the fact that its parent Hamiltonian can be block diagonalized and
related to a sawtooth lattice model. The analysis presented here may be
generalized to understand other non-Hermitian systems with multiple bands.
