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Astrocombs are ideal spectrograph calibrators whose limiting precision can be derived using a second, independent, astrocomb system. We therefore analyse data from two astrocombs (one 18 GHz and one 25 GHz) used simultaneously on the HARPS spectrograph at the European Southern Observatory. The first aim of this paper is to quantify the wavelength repeatability achieved by a particular astrocomb. The second aim is to measure wavelength calibration consistency between independent astrocombs, that is to place limits or measure any possible zero-point offsets. We present three main findings, each with important implications for exoplanet detection, varying fundamental constant and redshift drift measurements. Firstly, wavelength calibration procedures are important: using multiple segmented polynomials within one echelle order results in significantly better wavelength calibration compared to using a single higher-order polynomial. Segmented polynomials should be used in all applications aimed at precise spectral line position measurements. Secondly, we found that changing astrocombs causes significant zero-point offsets ($\approx 60{\rm cms}^{-1}$ in our raw data) which were removed. Thirdly, astrocombs achieve a precision of $\lesssim 4{\rm cms}^{-1}$ in a single exposure ($\approx 10\% $ above the measured photon-limited precision) and $1 {\rm cms}^{-1}$ when time-averaged over a few hours, confirming previous results. Astrocombs therefore provide the technological requirements necessary for detecting Earth-Sun analogues, measuring variations of fundamental constants and the redshift drift.