  Aims. We aim to investigate the ionisation conditions and distances of
Galactic high-velocity clouds (HVCs) in the Galactic halo and beyond in the
direction of the Local Group (LG) barycentre and anti-barycentre, by studying
spectral data of 29 extragalactic background sources obtained with Cosmic
Origins Spectropgraph (COS) installed on the Hubble Space Telescope (HST).
Methods. We model column-densities of low, intermediate, and high ions, such as
Si II, C II, Si III, Si IV, and C IV and use this to construct a set of Cloudy
ionisation models. Results. In total, we found 69 high-velocity absorption
components along the 29 lines of sight. The ones in the direction of the LG
barycentre span the entire range of studied velocities, 100 \lesssim
|v_{LSR}|\lesssim 400 km s^-1, while the anti-barycentre sample has velocities
up to about 300 km s^-1. For 49 components, we infer the gas densities. In the
direction of the LG barycentre, the gas densities exhibit a large range between
log n_H=-3.96 to -2.55, while in the anti-barycentre direction the densities
are systematically higher, log n_H>-3.25. The barycentre absorbers can be split
into two groups based on their density: a high density group with log
n_H>-3.54, which can be affected by the Milky Way radiation field, and a low
density group (log n_H \leq -3.54). The latter has very low thermal pressures
of P/k<7.3 K cm^-3. Conclusions. Our study shows that part of the absorbers in
the LG barycentre direction trace gas at very low gas densities and thermal
pressures. Such properties indicate that these absorbers are located beyond the
virial radius of the Milky Way. Our study also confirms results from earlier,
single-sightline studies, suggesting the presence of a metal-enriched
intragroup medium filling the LG near its barycentre.
