  The moduli space of principally polarized abelian varieties with real
structure and with level $N=4m$ structure (with $m \ge 1$) is shown to coincide
with the set of real points of a quasi-projective algebraic variety defined
over $\mathbb Q$, and to consist of finitely many copies of the quotient of the
space $\mathbf{GL}(n, \mathbb R)/\mathbf O(N)$ (of positive definite symmetric
matrices) by the principal congruence subgroup of level $N$ in $\mathbf{GL}(n,
\mathbb Z).$
