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For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$over $\mathbb{R}$, we give a criterion for the density in the parameter spaceof those abelian varieties over $\mathbb{R}$ containing a $k$-dimensionalabelian subvariety over $\mathbb{R}$. As application, we prove density of sucha set in the moduli space of polarized real abelian varieties of dimension $g$,and density of real algebraic curves mapping non-trivially to real$k$-dimensional abelian varieties in the moduli space of real algebraic curvesas well as in the space of real plane curves. This extends to the real settingresults by Colombo and Pirola as outlined in their paper "Some density resultsfor curves with non-simple jacobians", $\textit{Math. Ann.}$ (1990). We thenconsider the real locus of an algebraic stack over $\mathbb{R}$, attaching atopological space to it. For a real moduli stack, this defines a real modulispace. We show that for $\mathcal{M}_g$ and $\mathcal{A}_g$, the real modulispaces that arise in this way coincide with the moduli spaces of Gross-Harris("Real algebraic curves", $\textit{Ann. Sci. Ecole. Norm. Sup.}$ (1981)) andSepp\"al\"a-Silhol ("Moduli Spaces for Real Algebraic Curves and Real AbelianVarieties", $\textit{Math. Z.}$ (1989)).

Real moduli spaces and density of non-simple real abelian varieties