  We give a lattice-theoretic characterization for a manifold of
$\mathrm{OG}10$ type to be birational to some moduli space of (twisted) sheaves
on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of $\mathrm{OG}10$
type associated to any smooth cubic fourfold. Moreover we determine when a
birational transformation is induced by an automorphism of the K3 surface and
we use this to classify all induced birational symplectic involutions.
