We present a complete list of general relativistic shear-free solutions in a class of anisotropic, spatially homogeneous and orthogonal cosmological models containing a collection of n independent p-form gauge fields, where , in addition to standard ΛCDM matter fields modelled as perfect fluids. Here a (collection of) gauge field(s) balances anisotropic spatial curvature on the right-hand side of the shear propagation equation. The result is a class of solutions dynamically equivalent to standard FLRW cosmologies, with an effective curvature constant that depends both on spatial curvature and the energy density of the gauge field(s). In the case of a single gauge field (n = 1) we show that the only spacetimes that admit such solutions are the LRS Bianchi type III, Bianchi type VI0 and Kantowski–Sachs metric, which are dynamically equivalent to open (), flat () and closed () FLRW models, respectively. With a collection of gauge fields (n > 1) also Bianchi type II admits a shear-free solution (). We identify the LRS Bianchi type III solution to be the unique shear-free solution with a gauge field Hamiltonian bounded from below in the entire class of models.