  Spectroscopic surveys of the Galaxy reveal that its disc stars exhibit a
spread in $\mathrm{[\alpha/Fe]}$ at fixed $\mathrm{[Fe/H]}$, manifest at some
locations as a bimodality. The origin of these diverse, and possibly distinct,
stellar populations in the Galactic disc is not well understood. We examine the
Fe and $\alpha$-element evolution of 133 Milky Way-like galaxies from the EAGLE
simulation, to investigate the origin and diversity of their
$\mathrm{[\alpha/Fe]}$-$\mathrm{[Fe/H]}$ distributions. We find that bimodal
$\mathrm{[\alpha/Fe]}$ distributions arise in galaxies whose gas accretion
histories exhibit episodes of significant infall at both early and late times,
with the former fostering more intense star formation than the latter. The
shorter characteristic consumption timescale of gas accreted in the earlier
episode suppresses its enrichment with iron synthesised by Type Ia SNe,
resulting in the formation of a high-$\mathrm{[\alpha/Fe]}$ sequence. We find
that bimodality in $\mathrm{[\alpha/Fe]}$ similar to that seen in the Galaxy is
rare, appearing in approximately 5 percent of galaxies in our sample. We posit
that this is a consequence of an early gas accretion episode requiring the mass
accretion history of a galaxy's dark matter halo to exhibit a phase of
atypically-rapid growth at early epochs. The scarcity of EAGLE galaxies
exhibiting distinct sequences in the $\mathrm{[\alpha/Fe]}$-$\mathrm{[Fe/H]}$
plane may therefore indicate that the Milky Way's elemental abundance patterns,
and its accretion history, are not representative of the broader population of
$\sim L^\star$ disc galaxies.
