Measurements of a Higgs boson at the Large Hadron Collider (LHC) have become increasingly consistent with the predictions of the Standard Model (SM). This fact puts severe constraints on many potential low-energy extensions of the Higgs sector of the SM. In the well-known Two Higgs Doublet Model (2HDM), an `alignment limit' of parameters readily furnishes one SM-like scalar, and can be achieved naturally through an underlying symmetry. Among the other physical states of the 2HDM, a charged scalar H± would provide striking evidence of new physics if observed. We propose a novel technique for the observation of the process pp→tbH±→t¯tb¯b in the dileptonic decay channel at the LHC. The reconstruction of events in this channel is complicated by multiple b-jets and unobserved neutrinos in the final state. To determine the neutrino momenta, we implement a neutrino weighting procedure to study, for the first time, the t¯tb¯b signature. We further train a pair of boosted decision trees to reconstruct and classify signal events. We determine the resulting reach within the context of naturally aligned 2HDMs, such as the Maximally Symmetric Two Higgs Doublet Model (MS-2HDM). By testing at the integrated luminosity of 150 fb-1 achieved in Run 2 of the LHC, we find that this channel may restrict the parameter space of a Type-II MS-2HDM with charged Higgs masses as high as 680 GeV.