  For every $g\geq 2$ we distinguish real period matrices of real Riemann
surfaces of topological type $(g,0,0)$ from the ones of topological type
$(g,k,1)$, with $k$ equal to one or two for $g$ even or odd respectively
(Theorem B). To that purpose, we exhibit new invariants of real principally
polarized abelian varieties of orthosymmetric type (Theorem A.1). As a direct
application, we obtain an exhaustive criterion to decide about the existence of
real points on a real Riemann surface, requiring only a real period matrix of
its and the evaluation of the sign of at most one (real) theta constant
(Theorem C). A part of our real, algebro-geometric instruments first appeared
in the framework of nonlinear integrable partial differential equations.
