Measurements of tt¯H production in the H→bb¯ channel depend in a critical way on the theoretical uncertainty associated with the irreducible tt¯+b-jet background. In this paper, analysing the various topologies that account for b-jet production in association with a tt¯ pair, we demonstrate that the process at hand is largely driven by final-state g→bb¯ splittings. We also show that in five-flavour simulations based on tt¯+multi-jet merging, b-jet production is mostly driven by the parton shower, while matrix elements play only a marginal role in the description of g→bb¯ splittings. Based on these observations we advocate the use of NLOPS simulations of pp→tt¯bb¯ in the four-flavour scheme, and we present a new Powheg generator of this kind. Predictions and uncertainties for tt¯+b-jet observables at the 13 TeV LHC are presented both for the case of stable top quarks and with spin-correlated top decays. Besides QCD scale variations we consider also theoretical uncertainties related to the Powheg matching method and to the parton shower modelling, with emphasis on g→bb¯ splittings. In general, matching and shower uncertainties turn out to be remarkably small. This is confirmed also by a consistent comparison against Sherpa+OpenLoops.