  Convergence of the self-consistent skeleton diagrammatic technique (SDT) --
in which the full Green's function is determined through summation of Feynman
diagrams in terms of itself -- to the wrong answer has been associated with the
existence of non-perturbative branches of the Luttinger-Ward functional.
Although it has been possible to detect misleading convergence without the
knowledge of the exact result, the SDT has remained inapplicable in the regimes
where this happens. We show that misleading convergence does not always
preclude recovering the exact solution. In addition to the established
mechanism, convergence of the SDT to the wrong answer can stem from divergence
of the inherent diagrammatic series, which allows us to recover the exact
solution by a modified SDT protocol based on controlled analytic continuation.
We illustrate this approach by its application to the analytically solvable
(0+0)d Hubbard model, the Hubbard atom, and the 2d Hubbard model in a
challenging strong-coupling regime, for which the SDT is solved with controlled
accuracy by the diagrammatic Monte Carlo (DiagMC) technique.
