  We present a study of the intra-cluster population of low-mass X-ray binaries
(LMXB) residing in globular clusters (GC) in the central 1 $deg^2$ of the
Fornax galaxy cluster. Differently from previous studies, which were restricted
to the innermost regions of individual galaxies, this work is aimed at
comparing the properties of the intra-cluster population of GC-LMXBs with those
of the host galaxy. We use VLT Survey Telescope (VST) and Chandra observations.
We identify 168 LMXBs residing in GCs and divide it into host-galaxy and
intra-cluster objects based on their distance from the nearest galaxy in terms
of effective radius. We found 82 intra-cluster GC-LMXBs and 86 objects that are
hosted in galaxies. We perform a Gaussian mixture model to divide the
population into red and blue GCs. As has been found for the innermost regions
of galaxies, LMXBs tend to form in red and bright GCs in intra-cluster space as
well. We find, however, that the likelihood of a red GC to host an LMXB
decreases with galactocentric distance, but it remains approximately constant
for the blue GC population. Investigating the X-ray properties of the LMXBs
residing in GCs, we find a difference in the X-ray luminosity function between
the intra-cluster and host-galaxy sample: both follow a power-law down to $\sim
8.5\times 10^{37}$ erg s$^{-1}$, which is consistent with field LMXBs for the
intra-cluster sample, while the latter agree with previous estimates for LMXBs
in GCs. We detect a tentative difference in the hardness ratio of two
populations, where the intra-cluster GC-LMXBs appear to have harder spectra
than the host-galaxy objects. We find the same trend when we compare red and
blue GC-LMXBs: the spectra of the blue sample are harder spectra than those of
the red sample. This result could suggest a relation between the spectral
properties of LMXBs and the host GC colour and therefore its metallicity.
