Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed $2$-forms. It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $\mathbb{C}^{\infty}$-manifold.