  A condition of Osserman type, called $\phi$-null Osserman condition, is
introduced and studied in the context of Lorentz globally framed $f$-manifolds.
An explicit example shows the naturalness of this condition in the setting of
Lorentz $\mathcal{S}$-manifolds. We prove that a Lorentz $\mathcal{S}$-manifold
with constant $\phi$-sectional curvature is $\phi$-null Osserman, extending a
result stated for Lorentz Sasaki space forms. Then we state some
characterizations for a particular class of $\phi$-null Osserman
$\cal{S}$-manifolds. Finally, some examples are examined.
