  In this paper we prove the birational superrigidity of Fano-Mori fibre spaces
$\pi\colon V\to S$, every fibre of which is a complete intersection of type
$d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying
certain conditions of general position, under the assumption that the fibration
$V/S$ is sufficiently twisted over the base (in particular, under the
assumption that the $K$-condition holds). The condition of general position for
every fibre guarantees that the global log canonical threshold is equal to one.
This condition bounds the dimension of the base $S$ by a constant that depends
on the dimension $M$ of the fibre only (as the dimension $M$ of the fibre
grows, this constant grows as $\frac12 M^2$). The fibres and the variety $V$
itself may have quadratic and bi-quadratic singularities, the rank of which is
bounded from below.
