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We present a global time fit to determine the reference time event-by-event in an array of drift cells. The method is applied to the monitored drift tube tracking chambers (MDT) of the muon spectrometer of the ATLAS experiment at the large hadron collider (LHC). During operation with proton-proton collisions, the muon trigger is based on dedicated fast detectors and the readout is clocked at the beam crossing frequency of 40 MHz. A method has been developed for tracking particles through the MDT chambers which works without a precisely timed external trigger. This study was conceived to exploit the MDT tracking capabilities in non standard operation of the ATLAS detector, e.g., during commissioning with cosmic rays when there is no time correlation with the LHC clock frequency. Results obtained using an asynchronous trigger, with a high energy muon beam from the CERN SPS, and with cosmic rays are reported.