  We determine the rational real forms of the complex Mori fiber spaces for
which the identity component of the automorphism group is a maximal connected
algebraic subgroup of $\mathrm{Bir}(\mathbb{P}_{\mathbb{C}}^{3})$. This yields
a list of maximal connected algebraic subgroup of
$\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{3})$. We furthermore determine the
equivariant Sarkisov links starting from these rational real forms. This
article is the first step towards classifying all the maximal connected
algebraic subgroups of $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{3})$.
