  The Supergiant X-ray binary Vela X-1 represents one of the best astrophysical
sources to investigate the wind environment of a O/B star irradiated by an
accreting neutron star. Previous studies and hydrodynamic simulations of the
system revealed a clumpy environment and the presence of two wakes: an
accretion wake surrounding the compact object and a photoionisation wake
trailing it along the orbit. Our goal is to conduct, for the first time,
high-resolution spectroscopy on Chandra/HETG data at the orbital phase
$\varphi_\mathrm{orb} \approx 0.75$, when the line of sight is crossing the
photoionisation wake. We aim to conduct plasma diagnostics, inferring the
structure and the geometry of the wind. We perform a blind search employing a
Bayesian Block algorithm to find discrete spectral features and identify them
thanks to the most recent laboratory results or through atomic databases.
Plasma properties are inferred both with empirical techniques and with
photoionisation models within CLOUDY and SPEX. We detect and identify five
narrow radiative recombination continua (Mg XI-XII, Ne IX-X, O VIII) and
several emission lines from Fe, S, Si, Mg, Ne, Al, and Na, including four
He-like triplets (S XV, Si XIII, Mg XI, and Ne IX). Photoionisation models well
reproduce the overall spectrum, except for the near-neutral fluorescence lines
of Fe, S, and Si. We conclude that the plasma is mainly photoionised, but more
than one component is most likely present, consistent with a multi-phase plasma
scenario, where denser and colder clumps of matter are embedded in the hot,
photoionised wind of the companion star. Simulations with the future X-ray
satellites Athena and XRISM show that a few hundred seconds of exposure will be
sufficient to disentangle the lines of the Fe K$\alpha$ doublet and the He-like
Fe XXV, improving, in general, the determination of the plasma parameters.
