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We study twisted Jacobi manifolds, a concept that we had introduced in aprevious Note. Twisted Jacobi manifolds can be characterized using twistedDirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show thateach twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle.We introduce the notion of quasi-Jacobi bialgebroid and we prove that eachtwisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated.Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobialgebroid. Several examples of twisted Jacobi manifolds and twistedDirac-Jacobi structures are presented

Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids