  We study the transport properties of a diffusive Josephson junction between
two spin-split superconductors made of superconductor-ferromagnetic insulator
bilayers (FIS) on top of a 3D topological insulator (TI). We derive the
corresponding Usadel equation describing the quasiclassical Green's functions
in these systems and first solve the equation analytically in the
weak-proximity case. We demonstrate the appearance of an anomalous phase in the
absence of an external magnetic field. We also explore non-reciprocal
electronic transport. Specifically, we calculate the junction's diode
efficiency $\eta$ by solving the Usadel equation. We obtain a sizable diode
effect even at zero applied magnetic field. We discuss how the diode efficiency
$\eta$ depends on the different parameters and find a non-monotonic behavior of
$\eta$ with temperature.
