  We consider the pseudo-nearly K\"ahler
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ and we study its
Lagrangian submanifolds. We provide examples of Lagrangian submanifolds which
do not have an analogue in $\mathbb{S}^3\times\mathbb{S}^3$. We also provide an
expression for the isometry group of
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ with the
pseudo-Riemannian nearly K\"ahler metric. The main result is a complete
classification of extrinsically homogeneous Lagrangian submanifolds in this
space.
