In this paper, we study self-consistent solutions in one-dimensional lattice models obtained via many-body perturbation theory. The Dyson equation is solved in a fully self-consistent manner via the algorithmic-inversion method based on the sum-over-poles representation (AIM-SOP) of dynamical operators. In particular, we focus on the GW approximation, analyzing the spectral properties and the emergence of possible magnetic- or charge-density-wave broken symmetry solutions. We start by validating our self-consistent AIM-SOP implementation by taking as a test case the one-dimensional Hubbard model. We then move to the study of antiferromagnetic and charge-density-wave solutions in one-dimensional lattice models, taking into account a long-range Coulomb interaction between the electrons. We show that moving from local to nonlocal electronic interactions leads to a competition between antiferromagnetic and charge-density-wave broken symmetry solutions. Complementary, by solving the Sham-Schlüter equation, we can compute the noninteracting Green's function reproducing the same charge density of the interacting system. In turn, this allows for the evaluation of the derivative discontinuity of the Kohn-Sham potential, showing that its contribution to the fundamental gap can become dominating in some of the studied cases.