Given a family of holomorphic symplectic forms of a specific shape on a real manifold, a pseudo-hyper-Kähler structure exists directly on that manifold, and it agrees with the classical twistor construction when both apply.
Neitzke, `` Moduli of Higgs Bundles (Preliminary and incomplete draft) ,'' 2016
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A Concrete Variant of the Twistor Theorem
Given a family of holomorphic symplectic forms of a specific shape on a real manifold, a pseudo-hyper-Kähler structure exists directly on that manifold, and it agrees with the classical twistor construction when both apply.