  Accurate masses and radii for normal stars derived from observations of
detached eclipsing binary stars are of fundamental importance for testing
stellar models and may be useful for calibrating free parameters in these model
if the masses and radii are sufficiently precise and accurate. We aim to
measure precise masses and radii for the stars in the bright eclipsing binary
AI Phe, and to quantify the level of systematic error in these estimates. We
use several different methods to model the TESS light curve of AI Phe combined
with spectroscopic orbits from multiple sources to estimate precisely the
stellar masses and radii together with robust error estimates. We find that the
agreement between different methods for the light curve analysis is very good
but some methods underestimate the errors on the model parameters. The
semi-amplitudes of the spectroscopic orbits derived from spectra obtained with
modern echelle spectrographs are consistent to within 0.1%. The masses of the
stars in AI Phe are $M_1 = 1.1938 \pm 0.0008 M_{\odot}$ and $M_2 = 1.2438 \pm
0.0008M_{\odot}$, and the radii are $R_1 = 1.8050 \pm 0.0022 R_{\odot}$ and
$R_2 = 2.9332 \pm 0.0023 R_{\odot}$. We conclude that it is possible to measure
accurate masses and radii for stars in bright eclipsing binary stars to a
precision of 0.2% or better using photometry from TESS and spectroscopy
obtained with modern echelle spectrographs. We provide recommendations for
publishing masses and radii of eclipsing binary stars at this level of
precision.
