Without the constraint imposed by Hermiticity, non-Hermitian systems enjoy greater freedom than Hermitian ones. While the non-Hermitian ramification of non-spatial (internal) symmetries has been revealed, spatial symmetries remain to be explored. Here, we identify intrinsically non-Hermitian spatial symmetries using the same reasoning as non-spatial symmetry ramification. The symmetry endows exceptional topological semimetals with global topological structures, by preserving exceptional points but altering their topological invariants nonlocally. Furthermore, the global band configuration in the bulk is strongly constrained by non-Hermitian spatial symmetry, due to the intertwining of left and right eigen-systems at symmetry-related locations in momentum space. We illustrate our theory using two novel topological phases: exceptional unconventional Weyl semimetals and exceptional triple-point semimetals, in which the global structures of exceptional points, exceptional lines, and higher-order exceptional points are stabilized by one or more non-Hermitian spatial symmetries. We propose a cold-atom experiment to realize the exceptional unconventional Weyl semimetals.
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