  We study the emergence of non-Hermitian band topology in a two-dimensional
metal with planar spiral magnetism due to a momentum-dependent relaxation rate.
A sufficiently strong momentum dependence of the relaxation rate leads to
exceptional points in the Brillouin zone, where the Hamiltonian is
nondiagonalizable. The exceptional points appear in pairs with opposite
topological charges and are connected by arc-shaped branch cuts. We show that
exceptional points inside hole and electron pockets, which are generally
present in a spiral magnetic state with a small magnetic gap, can cause a
drastic change of the Fermi surface topology by merging those pockets at
isolated points in the Brillouin zone. We derive simple rules for the evolution
of the eigenstates under semiclassical motion through these crossing points,
which yield geometric phases depending only on the Fermi surface topology. The
spectral function observed in photoemission exhibits Fermi arcs. Its momentum
dependence is smooth -- despite of the nonanalyticities in the complex
quasiparticle band structure.
