  We study $3$-dimensional partially hyperbolic diffeomorphisms that are
homotopic to the identity, focusing on the geometry and dynamics of Burago and
Ivanov's center stable and center unstable \emph{branching} foliations. This
extends our study of the true foliations that appear in the dynamically
coherent case (see \emph{Partially hyperbolic diffeomorphisms homotopic to the
identity in dimension 3, Part I: The dynamically coherent case},
arxiv:1908.06227v3). We complete the classification of such diffeomorphisms in
Seifert fibered manifolds. In hyperbolic manifolds, we show that any such
diffeomorphism is either dynamically coherent and has a power that is a
discretized Anosov flow, or is of a new potential class called a \emph{double
translation}.
