of 7) 1604513PbHfS 3 , and SnHfS 3 . [2][3][4][5][6] Recent studies also revealed that iodate compounds, such as CH 3 NH 3 PbI 3 (which presents a high photovoltaic conversion) and CsSnI 3 can also crystallize in the NH 4 CdCl 3 -type structure. [7,8] Xu et al. further predicted, by using a first-principles-based genetic algorithm, that the NH 4 CdCl 3 -type structure can also become the ground state of many ABO 3 oxides and ABF 3 fluorites, when under high enough pressure. [9] Considering its generality and its occurrence after the perovskite (Pv) and post-perovskite (pPv) structures when gradually increasing the hydrostatic pressure in various ABO 3 and ABF 3 compounds, the NH 4 CdCl 3 -type structure is named as post-post-perovskite (ppPv) in ref.[9] -which is the convention we will also adopt in this paper.Interestingly, as far as we can tell, the post-post-perovskite structure has never been found (or predicted) to display an electrical polarization or the so-called Jahn-Teller (JT) distortions -intimately related to magnetism and electronic structure -in any material, including ABO 3 and ABF 3 systems. This is in line with the common belief that hydrostatic pressure suppresses structural distortions of the type that are typically related to electric polarization and Jahn-Teller effects. [10][11][12][13] For instance, experimental and computational works both reported that JT distortions of LaMnO 3 perovskite are reduced under pressure until potentially fully vanishing for a high enough compression. [11][12][13] Similarly, polarization typically results from a competition between long-range Coulomb interactions (which favor ferroelectricity) and short-range interactions (which favor the paraelectric structure), [14] and pressure tips this delicate balance in favor of short-range repulsive effects, thus favoring the paraelectric state. [15,16] Here we use first-principles simulations to predict that such beliefs have to be revisited. More precisely, we discover that applying pressure to many magnetic rare-earth manganites RMnO 3 not only leads to the stabilization of the post-postperovskite structure but also induces pseudo Jahn-Teller effects that, in turn, yield a spontaneous electrical polarization via a previously unknown coupling. Interestingly, the latter coupling only involves the polarization and the mentioned JT distortion; this feature sets the new coupling apart from other trilinear effects, [17,18] in which a third structural distortion appears, as it strongly suggests the possibility of an easier control of JT effects and related electronic properties via the application of Multiferroic materials are currently attracting a lot of interest because of the cross-coupling between their electrical and magnetic properties, which has the potential to lead to novel devices. There is a relatively small number of multiferroics and, hence, various strategies have been proposed to design materials possessing both ordered electric and magnetic dipoles. However, one strategy that has been scarcely investigated...