For sensitive infrared interferometry, it is crucial to control the differential piston evolution between the used telescopes. This is classically done by the use of a fringe tracker. In this work, we develop a new method to reconstruct the temporal piston variation from the atmosphere, by using real-time data from adaptive optics wavefront sensing: the Piston Reconstruction Experiment (P-REx). In order to understand the principle performance of the system in a realistic multilayer atmosphere it is first extensively tested in simulations. The gained insights are then used to apply P-REx to real data, in order to demonstrate the benefit of using P-REx as an auxiliary system in a real interferometer. All tests show positive results, which encourages further research and eventually a real implementation. Especially the tests on on-sky data showed that the atmosphere is, under decent observing conditions, sufficiently well structured and stable, in order to apply P-REx. It was possible to conveniently reconstruct the piston evolution in two-thirds of the datasets from good observing conditions (r$_0$ $\sim$ 30 cm). The main conclusion is that applying the piston reconstruction in a real system would reduce the piston variation from around 10 $\mu$m down to 1-2 $\mu$m over timescales of up to two seconds. This suggests an application for mid-infrared interferometry, for example for MATISSE at the VLTI or the LBTI. P-REx therefore provides the possibility to improve interferometric measurements without the need for more complex AO systems than already in regular use at 8m-class telescopes.