A real analogue of the Moore--Tachikawa category

For each complex semisimple group $G_{\mathbb{C}}$, Moore and Tachikawa conjectured the existence of a certain two-dimensional topological quantum field theory $\eta_{G_{\mathbb{C}}} : \mathrm{Cob}_2 \to \mathrm{MT}$ whose target category has complex Lie groups as objects and holomorphic symplectic varieties with Hamiltonian actions of the groups as morphisms. The conjecture is motivated by string theory, where $\eta_{G_{\mathbb{C}}}$ is obtained by taking the Higgs branch of some supersymmetric quantum field theories (called theories of class $S$) depending on $G_{\mathbb{C}}$ and a Riemann surface. The goal of this paper is to define a real analogue $\mathrm{MT}_{\mathbb{R}}$ of the target category $\mathrm{MT}$ and to rigorously prove that it is a category.