  We prove that every quasi-complete intersection ideal is obtained from a pair
of nested complete intersection ideals by way of a flat base change. As a
by-product we establish a rigidity statement for the minimal two-step Tate
complex associated to an ideal $I$ in a local ring $R$. Furthermore, we define
a minimal two-step complete Tate complex $T$ for each ideal $I$ in a local ring
$R$; and prove a rigidity result for it. The complex $T$ is exact if and only
if $I$ is a quasi-complete intersection ideal; and in this case, $T$ is the
minimal complete resolution of $R/I$ by free $R$-modules.
