  The bulk-boundary correspondence (BBC) relates in-gap boundary modes to bulk
topological invariants. In certain non-Hermitian topological systems,
conventional BBC becomes invalid in the presence of the non-Hermitian skin
effect (NHSE), which manifests as distinct energy spectra under the periodic
and open boundary conditions and massive eigenstate localization at boundaries.
In this work, we introduce a scheme to induce NHSE without breaking
conventional BBC, dubbed as the topologically compatible NHSE (TC-NHSE). In a
general one dimensional two-band model, we unveil two types of TC-NHSE that do
not alter topological phase transition points under any circumstance or only in
a certain parameter regime, respectively. Extending our model into two
dimension, we find that TC-NHSE can be selectively compatible to different sets
of Weyl points between different bands of the resultant semimetallic system,
turning some of them into bulk Fermi arcs while keeping the rest unchanged. Our
work hence helps clarify the intricate interplay between topology and NHSE in
non-Hermitian systems, and provides a versatile approach for designing
non-Hermitian topological systems where topological properties and NHSE do not
interfere each other.
