  We study non-Hermitian spatial symmetries -- a class of symmetries that have
no counterparts in Hermitian systems -- and study how normal and exceptional
semimetals can be stabilized by these symmetries. Different from internal ones,
spatial symmetries act nonlocally in momentum space and enforce global
constraints on both band degeneracies and topological quantities at different
locations. In deriving general constraints on band degeneracies and topological
invariants, we demonstrate that non-Hermitian spatial symmetries are on an
equal footing with, but are essentially different from Hermitian ones. First,
we discover the nonlocal Hermitian conjugate pair of exceptional or normal band
degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably,
we find that these pairs lead to the symmetry-enforced violation of the Fermion
doubling theorem in the long-time limit. Second, with the topological
constraints, we unravel that certain exceptional manifold is only compatible
with and stabilized by non-Hermitian spatial symmetries but is intrinsically
incompatible with Hermitian spatial symmetries. We illustrate these findings
using two three-dimensional models of a non-Hermitian Weyl semimetal and an
exceptional unconventional Weyl semimetal. Experimental cold-atom realizations
of both models are also proposed.
