We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism Hypothesis. This graphical calculus depends on a possibly singular foliation, and we use it to construct twisted string net modules for surfaces equipped with a Morse function or a Morse foliation. We prove that, despite the apparent dependence on this Morse function, the twisted string net modules assemble in an oriented categorified 2-TQFT. We study when the twisted string net module of the 2-sphere vanishes, relate it to the distinguished invertible object for finite tensor categories and exhibit examples of non-unimodular finite tensor categories with non-vanishing twisted string net module on the 2-sphere. This vanishing is expected to be the main obstruction for extending our categorified 2-TQFT to a non-compact 3-TQFT.
Haïoun et al. (Wed,) studied this question.