  Globular clusters (GCs) are bright objects that span a wide range of
galactocentric distances, and are thus probes of the structure of dark matter
(DM) haloes. In this work, we explore whether the projected radial profiles of
GCs can be used to infer the structural properties of their host DM haloes. We
use the simulated GC populations in a sample of 166 central galaxies from the
$(34.4~\rm cMpc)^3$ periodic volume of the E-MOSAICS project. We find that more
massive galaxies host stellar and GC populations with shallower density
profiles that are more radially extended. In addition, the metal-poor GC
subpopulations tend to have shallower and more extended profiles than the
metal-rich subsamples, which we relate to the preferentially accreted origin of
the metal-poor GCs. We find strong correlations between the slopes and
effective radii of the radial profiles of the GC populations and the structural
properties of the DM haloes, such as their power-law slopes, scale radii, and
concentration parameters. Accounting for a dependence on the galaxy stellar
mass decreases the scatter of the two-dimensional relations. This suggests that
the projected number counts of GCs, combined with their galaxy mass, trace the
density profile of the DM halo of their host galaxy. When applied to
extragalactic GC systems, we recover the scale radii and the extent of the DM
haloes of a sample of ETGs with uncertainties smaller than $0.2~\rm dex$. Thus,
extragalactic GC systems provide a novel avenue to explore the structure of DM
haloes beyond the Local Group.
