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Dynamical screening is a key property of charged many-particle systems. Its theoretical description is based on the $GW$ approximation that is extensively applied for ground-state and equilibrium situations but also for systems driven out of equilibrium. The main limitation of the $GW$ approximation is the neglect of strong electronic correlation effects that are important in many materials as well as in dense plasmas. Here we derive the dynamically screened ladder (DSL) approximation that selfconsistently includes, in addition to the $GW$ diagrams, also particle--particle and particle--hole $T$-matrix diagrams. The derivation is based on reduced-density-operator theory and the result is equivalent to the recently presented G1--G2 scheme [Schlünzen \textit{et al.}, Phys. Rev. Lett. \textbf{124}, 076601 (2020); Joost \textit{et al.}, Phys. Rev. B \textbf{101}, 245101 (2020)]. We perform extensive time-dependent DSL simulations for finite Hubbard clusters and present tests against exact results that confirm excellent accuracy as well as total energy conservation of the approximation. At strong coupling and for long simulation durations, instabilities are observed. These problems are solved by enforcing contraction consistency and applying a purification approach.