  We give a signed generalization of Laurent's theorem that characterizes
feasible positive semidefinite matrix completion problems in terms of metric
polytopes. Based on this result, we give a characterization of the maximum rank
completions of the signed positive semidefinite matrix completion problem for
odd-$K_4$ minor free signed graphs. The analysis can also be used to bound the
minimum rank over the completions and to characterize uniquely solvable
completion problems for odd-$K_4$ minor free signed graphs. As a corollary we
derive a characterization of the universal rigidity of odd-$K_4$ minor free
spherical tensegrities, and also a characterization of signed graphs whose
signed Colin de Verdi\`ere parameter $\nu$ is bounded by two, recently shown by
Arav et al.
