  We classify real hypersurfaces with isometric Reeb flow in the complex
hyperbolic quadrics ${Q^*}^{m} = SO^{o}_{2,m}/SO_mSO_2$, $m \geq 3$. We show
that $m$ is even, say $m = 2k$, and any such hypersurface becomes an open part
of a tube around a $k$-dimensional complex hyperbolic space ${\mathbb C}H^k$
which is embedded canonically in ${Q^*}^{2k}$ as a totally geodesic complex
submanifold or a horosphere whose center at infinity is $\frak A$-isotropic
singular. As a consequence of the result, we get the non-existence of real
hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics
${Q^*}^{2k+1}$, $k \geq 1$.
