  We introduce two simplicial complexes, the noncrossing matching complex and
the noncrossing bipartite complex. Both complexes are intimately related to the
bubble lattice introduced in our earlier article "Bubble Lattices I: Structure"
(arXiv:2202.02874). We study these complexes from both an enumerative and a
geometric point of view. In particular, we prove that these complexes are
shellable and give explicit formulas for certain refined face numbers. Lastly,
we conjecture an intriguing connection of these refined face numbers to the
so-called M-triangle of the shuffle lattice.
