Rationally connected threefolds with nef and bad anticanonical divisor, II

Let [Formula: see text] be a smooth complex projective rationally connected threefold with [Formula: see text] being nef and not semi-ample. In our previous work, we classified all such threefolds when [Formula: see text] has no fixed divisor. In this paper, we continue our classification when [Formula: see text] has a non-zero fixed divisor. We show that the mobile part of [Formula: see text] can be non-rational, and in this case a general member of the mobile part of [Formula: see text] has exactly two irreducible components. We give a complete classification in this case and we also describe the cones of curves of the threefolds appearing in the classification.