In an earlier paper, we studied manifolds M endowed with a generalized F structure Φ∈End(TM⊕T∗M)\documentclass[12pt]{minimal}
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\begin{document}$$\Phi \in \mathrm{End}(TM\oplus T^*M)$$\end{document}, skew-symmetric with respect to the pairing metric, such that Φ3+Φ=0\documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^3+\Phi =0$$\end{document}. Furthermore, if Φ\documentclass[12pt]{minimal}
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\begin{document}$$\Phi $$\end{document} is integrable (in some well-defined sense), Φ\documentclass[12pt]{minimal}
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\begin{document}$$\Phi $$\end{document} is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields (A∈End(TM),π∈∧2TM)\documentclass[12pt]{minimal}
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\begin{document}$$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$\end{document}, where A3+A=0\documentclass[12pt]{minimal}
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\begin{document}$$A^3+A=0$$\end{document} and some relations between A and π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} hold. We establish the integrability conditions in terms of (A,π)\documentclass[12pt]{minimal}
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\begin{document}$$(A,\pi )$$\end{document}. They include the facts that A is a classical CRF structure, π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} is a Poisson bivector field and imA\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{im}\,A$$\end{document} is a (non)holonomic Poisson submanifold of (M,π)\documentclass[12pt]{minimal}
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\begin{document}$$(M,\pi )$$\end{document}. We discuss the case where either kerA\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{ker}\,A$$\end{document} or imA\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{im}\,A$$\end{document} is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of imA\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{im}\,A$$\end{document} inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}, including an associated spectral sequence and a Dolbeault type grading.
