Real forms of Mori fiber spaces with many symmetries

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 $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{3})$.