  The starting point of this note is our recent paper with Laza and Sacc\`a on
the construction of deformations of O'Grady's $10$-dimensional manifolds as
compactifications of intermediate Jacobian fibrations associated to cubic
fourfolds. The note provides a complement to that paper consisting in the
analogous construction in the twisted case, leading to isogenous but presumably
not isomorphic or birational hyper-K\"ahler manifolds.
