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The power conversion efficiencies (PCEs) of organic solar cells (OSCs) using non-fullerene acceptors (NFAs) have now reached 18%. However, this is still lower than inorganic solar cells, for which PCEs >20% are commonplace. A key reason is that OSCs still show low open-circuit voltages (Voc) relative to their optical band gaps, attributed to non-radiative recombination. For OSCs to compete with inorganics in efficiency, all non-radiative loss pathways must be identified and where possible, removed. Here, we show that in most NFA OSCs, the majority of charge recombination at open-circuit proceeds via formation of non-emissive NFA triplet excitons (T1); in the benchmark PM6:Y6 blend, this fraction reaches 90%, contributing 60 mV to the reduction of Voc. We develop a new design to prevent recombination via this non-radiative channel through the engineering of significant hybridisation between the NFA T1 and the spin-triplet charge transfer exciton (3CTE). We model that the rate of the back charge transfer from 3CTE to T1 can be reduced by an order of magnitude, allowing re-dissociation of the 3CTE. We then demonstrate NFA systems where T1 formation is suppressed. This work therefore provides a clear design pathway for improved OSC performance to 20% PCE and beyond.