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The freely available Monte Carlo code REAS for modelling radio emission from cosmic ray air showers has evolved to include the full complexity of air shower physics. REAS3 improves the calculation of the emission contributions, which was not fully consistent in earlier versions of REAS, by incorporating the missing radio emission due to the variation of the number of charged particles during the air shower evolution using an "end-point formalism". With the inclusion of these emission contributions, the structure of the simulated radio pulses changes from unipolar to bipolar, and the azimuthal emission pattern becomes nearly symmetric. Remaining asymmetries can be explained by radio emission due to the variation of the net charge excess in air showers, which is automatically taken into account in the new implementation. REAS3 constitutes the first self-consistent time-domain implementation based on single particle emission taking the full complexity of air shower physics into account, and is freely available for all interested users. REAS3 has been superseded by CoREAS (ascl:1406.003).