We study the formation of planes of dwarf galaxies around Milky Way (MW)-mass haloes in the EAGLE galaxy formation simulation. We focus on satellite systems similar to the one in the MW: spatially thin or with a large fraction of members orbiting in the same plane. To characterise the latter, we introduce a robust method to identify the subsets of satellites that have the most co-planar orbits. Out of the 11 MW classical dwarf satellites, 8 have highly clustered orbital planes whose poles are contained within a $22^circ$ opening angle centred around $(l,b)=(182^circ,-2^circ)$. This configuration stands out when compared to both isotropic and typical $Lambda$CDM satellite distributions. Purely flattened satellite systems are short-lived chance associations and persist for less than $1~rm{Gyr}$. In contrast, satellite subsets that share roughly the same orbital plane are longer lived, with half of the MW-like systems being at least $4~rm{Gyrs}$ old. On average, satellite systems were flatter in the past, with a minimum in their minor-to-major axes ratio about $9~rm{Gyrs}$ ago, which is the typical infall time of the classical satellites. MW-like satellite distributions have on average always been flatter than the overall population of satellites in MW-mass haloes and, in particular, they correspond to systems with a high degree of anisotropic accretion of satellites. We also show that torques induced by the aspherical mass distribution of the host halo channel some satellite orbits into the hosts equatorial plane, enhancing the fraction of satellites with co-planar orbits. In fact, the orbital poles of co-planar satellites are tightly aligned with the minor axis of the host halo.
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