  The inclusion of supersymmetric threshold corrections to the $b$-quark mass
has dramatic consequences in scenarios with large CP-mixing effects in the
Higgs sector. In particular, when the phase of the combination
$M_{\tilde{g}}\mu$ is $\sim 180^{\rm o} \pm 30^{\rm o}$, the lightest-sbottom
squark becomes tachyonic and, possibly, the $b$-quark Yukawa coupling
nonperturbative for values of $\tan \beta$ ranging from intermediate up to
large or very large, depending on the size of $arg(A_t\mu)$, $arg(A_b\mu)$, and
the details of the spectrum. For these scenarios, when allowed, as well as
scenarios with different values of $arg(M_{\tilde{g}}\mu)$, the cross sections
for the production of the three neutral-Higgs bosons through $b$-quark have
rather interesting dependences on $arg(A_t\mu)$ and $arg(A_b\mu)$, and the
deviations induced by the $m_b$ corrections are rather large. In general, such
production channel cannot be neglected with respect to the production through
gluon fusion. For large CP-mixing effects, the lightest neutral-Higgs boson can
be mainly CP odd and the $b$-quark fusion becomes its main production
mechanism. Searches at the Tevatron and the LHC can easily detect such a Higgs
boson, or constrain the CP-violating scenarios that allow it.
