  We introduce the notion of Reeb parallel structure Jacobi operator for real
hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$,
$m \geq 3$, and give a classification theory for real hypersurfaces in
${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.
