  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
$p\in\{0,1,2,3\}$, in addition to standard $\Lambda$CDM 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 $K_{\rm{eff}}$ 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 VI$_0$ and
Kantowski-Sachs metric, which are dynamically equivalent to open
($K_{\rm{eff}}<0$), flat ($K_{\rm{eff}}=0$) and closed ($K_{\rm{eff}}>0$) FLRW
models, respectively. With a collection of gauge fields ($n>1$) also Bianchi
type II admits a shear-free solution ($K_{\rm{eff}}>0$). 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.
