Measurements of tt¯H production in the H→bb¯ channel depend in a critical way on the theoretical uncertainty associated with the irreducible QCD tt¯+b-jet background. We introduce a new pp→tt¯bb¯ POWHEG generator in the 4F scheme based on POWHEG-BOX-RES and on OpenLoops for fast evaluation of the scattering amplitudes. We present predictions and uncertainties for tt¯+b-jet observables at the 13 TeV LHC. We also consider theoretical uncertainties related to the POWHEG matching method and to the parton shower (PS) modelling, with emphasis on g→bb¯ splittings. In general, matching and shower uncertainties turn out to be remarkably small. This is confirmed by a consistent comparison against SHERPA+OpenLoops.