  In this paper, we study Lagrangian submanifolds of the pseudo-nearly K\"ahler
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$. First, we show that
they split into four different classes depending on their behaviour with
respect to a certain almost product structure on the ambient space. Then, we
give a complete classification of totally geodesic Lagrangian submanifolds of
this space.
