  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
host's 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.
