  The CSP dichotomy conjecture has been recently established, but a number of
other dichotomy questions remain open, including the dichotomy classification
of list homomorphism problems for signed graphs. Signed graphs arise naturally
in many contexts, including for instance nowhere-zero flows for graphs embedded
in non-orientable surfaces. For a fixed signed graph $\widehat{H}$, the list
homomorphism problem asks whether an input signed graph $\widehat{G}$ with
lists $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G}),$ admits a
homomorphism $f$ to $\widehat{H}$ with all $f(v) \in L(v), v \in
V(\widehat{G})$. Usually, a dichotomy classification is easier to obtain for
list homomorphisms than for homomorphisms, but in the context of signed graphs
a structural classification of the complexity of list homomorphism problems has
not even been conjectured, even though the classification of the complexity of
homomorphism problems is known. Kim and Siggers have conjectured a structural
classification in the special case of ``weakly balanced" signed graphs. We
confirm their conjecture for reflexive and irreflexive signed graphs; this
generalizes previous results on weakly balanced signed trees, and weakly
balanced separable signed graphs \cite{separable,trees}. In the reflexive case,
the result was first presented in \cite{KS}, with the proof using some of our
results included in this paper. In fact, here we present our full proof, as an
alternative to the proof in \cite{KS}. In particular, we provide direct
polynomial algorithms where previously algorithms relied on general dichotomy
theorems. The irreflexive results are new, and their proof depends on first
deriving a theorem on extensions of min orderings of (unsigned) bipartite
graphs, which is interesting on its own. [shortened, full abstract in PDF]
