An 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 paper, we generalize this notion to the 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 nontrivial 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.