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Given a Lagrangian fibration $\pi : X \to \mathbb{P}^n$ of a compacthyper-K\"ahler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or$\text{OG6}$-type, we construct a natural compactification of its dual torusfibration. Specifically, this compactification is given by a quotient of $X$ bycertain automorphisms acting trivially on the second cohomology and respectingthe Lagrangian fibration. It is a compact hyper-K\"ahler orbifold withidentical period mapping behavior as $X$.

The dual Lagrangian fibration of known hyper-K\"ahler manifolds