  We show that multichannel interactions significantly alter the phase diagram
of ultracold bosonic molecules in an optical lattice. Most prominently, an
unusual fluid region intervenes between the conventional superfluid and the
Mott insulator. In it, number fluctuations remain but phase coherence is
suppressed by a significant factor. This factor can be made arbitrarily large,
at least in a two-site configuration. We calculate the phase diagram using
complementary methods, including Gutzwiller mean-field and density matrix
renormalization group (DMRG) calculations. Although we focus on bosonic
molecules without dipolar interactions, we expect multichannel interactions to
remain important for dipolar interacting and fermionic molecules.
