A condition of Osserman type, called the φ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturality of this condition in the setting of Lorentz 𝓢-manifolds. We prove that a Lorentz 𝓢-manifold with constant φ-sectional curvature is φ-null Osserman, extending a well-known result in the case of Lorentz Sasaki space forms. Then we state a characterization of a particular class of φ-null Osserman 𝓢-manifolds. Finally, some examples are examined.