State-of-the-art lattice QCD studies of hot and dense strongly interacting matter currently rely on extrapolation from zero or imaginary chemical potentials. The ill-posedness of numerical analytic continuation puts severe limitations on the reliability of such methods. Here we use the more direct sign reweighting method to perform lattice QCD simulation of the QCD chiral transition at finite real baryon density on phenomenologically relevant lattices. This method does not require analytic continuation and avoids the overlap problem associated with generic reweighting schemes, so has only statistical but no uncontrolled systematic uncertainties for a fixed lattice setup. This opens up a new window to study hot and dense strongly interacting matter from first principles. We perform simulations up to a baryochemical potential-temperature ratio of $mu_B/T=2.5$ covering most of the RHIC Beam Energy Scan range in the chemical potential. We also clarify the connection of the approach to the more traditional phase reweighting method.
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