Towards a modular construction of OG10

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-K\"ahler manifold of $\mathrm{K3}^{[2]}$-type whose deformation space is smooth of dimension ten. Its moduli space is birational to a hyper-K\"ahler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-K\"ahler could lead to new examples of hyper-K\"ahler manifolds.