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 ${\eta }_{G_{\mathbb{C}}}:{\mathrm{C}\mathrm{o}\mathrm{b}}_2\to \mathrm{M}\mathrm{T}$ 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 ${\mathrm{M}\mathrm{T}}_{\mathbb{R}}$ of the target category $\mathrm{M}\mathrm{T}$ and to rigorously prove that it is a category.