  We introduce a contact invariant in the bordered sutured Heegaard Floer
homology of a three-manifold with boundary. The input for the invariant is a
contact manifold $(M, \xi, \mathcal{F})$ whose convex boundary is equipped with
a signed singular foliation $\mathcal{F}$ closely related to the characteristic
foliation. Such a manifold admits a family of foliated open book decompositions
classified by a Giroux Correspondence, as described in earlier work of Licata
and V\'ertesi. We use a special class of foliated open books to construct
admissible bordered sutured Heegaard diagrams and identify well-defined classes
$c_D$ and $c_A$ in the corresponding bordered sutured modules.
  Foliated open books exhibit user-friendly gluing behavior, and we show that
the pairing on invariants induced by gluing compatible foliated open books
recovers the Heegaard Floer contact invariant for closed contact manifolds. We
also consider a natural map associated to forgetting the foliation
$\mathcal{F}$ in favor of the dividing set, and show that it maps the bordered
sutured invariant to the contact invariant of a sutured manifold defined by
Honda-Kazez-Mati\'{c}.
