  We consider the effect of the internal nuclear quadrupole interaction on
quantum tunneling in complex multi-atomic two-level systems. Two distinct
regimes of strong and weak interactions are found. The regimes depend on the
relationship between a characteristic energy of the nuclear quadrupole
interaction $\lambda_{\ast}$ and a bare tunneling coupling strength
$\Delta_{0}$. When $\Delta_{0}>\lambda_{\ast}$, the internal interaction is
negligible and tunneling remains coherent determined by $\Delta_{0}$. When
$\Delta_{0}<\lambda_{\ast}$, coherent tunneling breaks down and an effective
tunneling amplitude decreases by an exponentially small overlap factor
$\eta^{\ast}\ll1$ between internal ground states of left and right wells of a
tunneling system. This affects thermal and kinetic properties of tunneling
systems at low temperatures $T<\lambda_{*}$. The theory is applied for
interpreting the anomalous behavior of the resonant dielectric susceptibility
in amorphous solids at low temperatures $T\leq 5$mK where the nuclear
quadrupole interaction breaks down coherent tunneling. We suggest the
experiments with external magnetic fields to test our predictions and to
clarify the internal structure of tunneling systems in amorphous solids.
