  We numerically study models of $\mathrm{H}_2$ and $\mathrm{LiH}$ molecules,
aligned collinearly with the linear polarization of the external field, to
elucidate the possible role of correlation in the enhanced-ionization
phenomena. Correlation is considered at different levels of approximation with
the time-dependent generalized-active-space configuration-interaction method.
The results of our studies show that enhanced ionization occurs in
multielectron molecules, that correlation is important and they also
demonstrate significant deviations between the results of the
single-active-electron approximation and more accurate
configuration-interaction methods. With the inclusion of correlation we show
strong carrier-envelope-phase effects in the enhanced ionization of the
asymmetric heteronuclear $\mathrm{LiH}$-like molecule. The correlated
calculation shows an intriguing feature of cross-over in enhanced ionization
with two carrier-envelope-phases at critical inter-nuclear separation.
