In this work, an alternative approach to data-driven surrogate models is explored for the microscale analysis of unidirectional laminates with rate-dependency. Recently, Recurrent Neural Networks (RNNs) have gained traction in material modeling for their generality in dealing with sequential data and for reducing the computational cost involved in multiscale simulations. In that regard, having an efficient microscale analysis is especially beneficial when techniques such as FE2 are considered. In this techique, a finite element model representing the geometry of the material at the lower scale (the Representative Volume Element (RVE)) is nested to each integration point at the macroscale and both scales are solved concurrently. However, the limited ability of the purely data-driven surrogate models to extrapolate outside their training spaces makes their reliable use in material modeling to remain a challenging endeavor. Moreover, RNNs are still severely limited by the curse of dimensionality associated with sampling arbitrarily long strain paths. The alternative explored here to circumvent those issues follows from the developments in [1], where we reintroduce some of the physics-based knowledge of the problem into the network by employing the actual material models used in the full-order micromodel inside the data-driven model. This time, an architecture suitable for finite strains framework is explored, in which the homogenized deformation gradient of the RVE is encoded into a set of deformation gradients corresponding to fictitious material points that are evaluated by the material models. The resulting stresses are then combined to obtain the homogenized stresses. Since every material point in the layer in which the material models are introduced keeps track of its own internal variables, rate-dependency arises naturally. The novel approach is employed as the surrogate model for a unidirectional composite RVE with transversely isotropic elastic fibers and elasto-viscoplastic matrix material. The performance is assessed in a set of loading cases where the RVE is subjected to different off-axis loading and strain-rates [2], including creep behavior. A key outcome of the proposed physically recurrent neural network is its ability to predict the homogenized response of the RVE to strain-rates and off-axis angles never seen during training.