  We describe a sequence of smooth quotients of the Deligne-Mumford moduli
space ${\mathbb R}\overline{\mathcal M}_{0,\ell+1}$ of real rational curves
with $\ell\!+\!1$ conjugate pairs of marked points that terminates at ${\mathbb
R}\overline{\mathcal M}_{0,\ell}\!\times\!{\mathbb C}{\mathbb P}^1$. This
produces an analogue of Keel's blowup construction of the Deligne-Mumford
moduli spaces $\overline{\mathcal M}_{\ell+1}$ of rational curves with
$\ell\!+\!1$ marked points, but with an explicit description of the
intermediate spaces and the blowups of three different types. The same
framework readily adapts to the real moduli spaces with real points. In a
sequel, we use this inductive construction of ${\mathbb R}\overline{\mathcal
M}_{0,\ell+1}$ to completely determine the rational (co)homology ring of
${\mathbb R}\overline{\mathcal M}_{0,\ell}$.
