  The complex eigenenergies and non-orthogonal eigenstates of non-Hermitian
systems exhibit unique topological phenomena that cannot appear in Hermitian
systems. Representative examples are the non-Hermitian skin effect and
exceptional points. In a two-dimensional parameter space, topological
classifications of non-separable bands in multiband non-Hermitian systems can
be established by invoking a permutation group, where the product of the
permutation represents state exchange due to exceptional points in the space.
We unveil in this work the role of pseudo-Hermitian lines in non-Hermitian
topology for multiple bands. Contrary to current understanding, the
non-separability of non-Hermitian multibands can be topologically non-trivial
without exceptional points in two-dimensional space. Our work builds on the
fundamental and comprehensive understanding of non-Hermitian multiband systems
and also offers versatile applications and realizations of non-Hermitian
systems without the need to consider exceptional points.
