We present an improved method for the precise reconstruction of cosmic ray air showers above $10^{17}$ eV with sparse radio arrays. The method is based on the comparison of predictions for radio pulse shapes by CoREAS simulations to measured pulses. We applied our method to the data of Tunka-Rex, a 1 km$^2$ radio array in Siberia operating in the frequency band of 30-80 MHz. Tunka-Rex is triggered by the air-Cherenkov detector Tunka-133 and by scintillators (Tunka-Grande). The instrument collects air-shower data since 2012. The present paper describes updated data and signal analyses of Tunka-Rex and details of a new method applied. After efficiency cuts, when Tunka-Rex reaches its full efficiency, the energy resolution of about 10% given by the new method has reached the limit of systematic uncertainties due to the calibration uncertainty and shower-to-shower fluctuations. At the same time the shower maximum reconstruction is significantly improved up to an accuracy of 35 g/cm$^2$ compared to the previous method based on the slope of the lateral distribution. We also define and now achieved conditions of the measurements, at which the shower maximum resolution of Tunka-Rex reaches a value of 25 g/cm$^2$ and becomes competitive to optical detectors. To check and validate our reconstruction and efficiency cuts we compare individual events to the reconstruction of Tunka-133. Furthermore, we compare the mean of shower maximum as a function of primary energy to the measurements of other experiments.
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