  Alice string is a topological defect with a very peculiar feature. When a
defect with a monopole charge encircles an Alice string, the monopole charge
changes sign. In this work, we generalize this notion to momentum space of
periodic media with loss and gain. In particular, we find that the generic
band-structure node for a three-dimensional non-Hermitian crystalline system
acts as an Alice string, which can flip the Chern number charge carried by Weyl
points and by exceptional-line rings. We discuss signatures of this topological
structure for a lattice model with one tuning parameter, including non-trivial
braiding of bulk band nodes, and the spectroscopic features of both the bulk
and the surface states. We also explore how an Alice string affects the
validity of the Nielsen-Ninomiya theorem, and present a mathematical
description of the braiding phenomenon.
