We present a Monte Carlo generator that implements significant theoretical improvements in the simulation of top-quark pair production and decay at the LHC. Spin correlations and off-shell effects in top-decay chains are described in terms of exact matrix elements for pp → l+νl l−ν ̄l b ̄b at NLO QCD, where the leptons l and l belong to different families, and b quarks are massive. Thus, the contributions from tt ̄ and W t single-top production as well as their quantum interference are fully included. Matrix elements are matched to the Pythia8 parton shower using a recently proposed method that allows for a consistent treatment of resonances in the POWHEG framework. These theoretical improve- ments are especially important for the interpretation of precision measurements of the top-quark mass, for single-top analyses in the W t channel, and for tt ̄ and W t backgrounds in the presence of jet vetoes or cuts that enhance off-shell effects. The new generator is based on a process-independent interface of the OpenLoops amplitude generator with the POWHEG-BOX framework.