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Hybrid AMX₃ perovskites (A = Cs, CH₃NH₃; M = Sn, Pb; X = halide) have revolutionized the scenario of emerging photovoltaic technologies, with very recent results demonstrating 15% efficient solar cells. The CH₃NH₃PbI₃/MAPb(I_1−xCl_x)₃ perovskites have dominated the field, while the similar CH₃NH₃SnI₃ has not been exploited for photovoltaic applications. Replacement of Pb by Sn would facilitate the large uptake of perovskite-based photovoltaics. Despite the extremely fast progress, the materials electronic properties which are key to the photovoltaic performance are relatively little understood. Density Functional Theory electronic structure methods have so far delivered an unbalanced description of Pb- and Sn-based perovskites. Here we develop an effective GW method incorporating spin-orbit coupling which allows us to accurately model the electronic, optical and transport properties of CH₃NH₃SnI₃ and …