The Magellanic Clouds are very rare in the IllustrisTNG simulations

The Large and Small Magellanic Cloud (LMC and SMC) form the closest interacting galactic system to the Milky Way, therewith providing a laboratory to test cosmological models in the local Universe. We quantify the likelihood for the Magellanic Clouds (MCs) to be observed within the $\Lambda$CDM model using hydrodynamical simulations of the IllustrisTNG project. The orbits of the MCs are constrained by proper motion measurements taken by the $Hubble~Space~Telescope$ and $Gaia$. The MCs have a mutual separation of $d_{\mathrm{MCs}}~=~24.5\,\mathrm{kpc}$ and a relative velocity of $v_{\mathrm{MCs}}~=~90.8\,\mathrm{km\,s^{-1}}$, implying a phase-space density of $f_{\mathrm{MCs,obs}}~\equiv~(d_{\mathrm{MCs}} \cdot v_{\mathrm{MCs}})^{-3}~=~9.10\times10^{-11}\,\mathrm{km^{-3}\,s^{3}\,kpc^{-3}}$. We select analogues to the MCs based on their stellar masses and distances in MW-like halos. None of the selected LMC analogues have a higher total mass and lower Galactocentric distance than the LMC, resulting in $>3.75\sigma$ tension. We also find that the $f_{\mathrm{MCs}}$ distribution in the highest resolution TNG50 simulation is in $3.95\sigma$ tension with observations. Thus, a hierarchical clustering of two massive satellites like the MCs in a narrow phase-space volume is unlikely in $\Lambda$CDM, presumably because of short merger timescales due to dynamical friction between the overlapping dark matter halos. We show that group infall led by an LMC analogue cannot populate the Galactic disc of satellites (DoS), implying that the DoS and the MCs formed in physically unrelated ways in $\Lambda$CDM. Since the $20^\circ$ alignment of the LMC and DoS orbital poles has a likelihood of $P=0.030$ ($2.17\sigma$), adding this $\chi^2$ to that of $f_{\mathrm{MCs}}$ gives a combined likelihood of $P = 3.90\times10^{-5}$ ($4.11\sigma$).