  Spectra of bulk or edges in topological insulators are often made complex by
non-Hermiticity. Here, we show that symmetry protection enables entirely real
spectra for both bulk and edges even in non-Hermitian topological insulators.
In particular, we demonstrate entirely real spectra without non-Hermitian skin
effects due to a combination of pseudo-Hermiticity and Kramers degeneracy. This
protection relies on nonspatial fundamental symmetry and has stability against
disorder. As an illustrative example, we investigate a non-Hermitian extension
of the Bernevig-Hughes-Zhang model. The helical edge states exhibit oscillatory
dynamics due to their nonorthogonality as a unique non-Hermitian feature.
