  Context: Massive colliding-wind binaries (CWBs) can be non-thermal sources.
The emission produced in their wind-collision region (WCR) encodes information
of both the shocks properties and the relativistic electrons accelerated in
them. The recently discovered system Apep, a unique massive system hosting two
Wolf-Rayet stars, is the most powerful synchrotron radio emitter among the
known CWBs, being an exciting candidate to investigate the non-thermal
processes associated with stellar wind shocks.
  Aims: We intend to break the degeneracy between the relativistic particle
population and the magnetic field strength in the WCR of Apep by probing its
hard X-ray spectrum, where inverse-Compton (IC) emission is expected to
dominate.
  Methods: We observe Apep with NuSTAR for 60 ks and combine this with a
re-analysis of a deep archival XMM-Newton observation to better constrain the
X-ray spectrum. We use a non-thermal emission model to derive physical
parameters from the results.
  Results: We detect hard X-ray emission consistent with a power-law component.
This is compatible with IC emission produced in the WCR for a magnetic field of
100-160 mG and a fraction of ~1.5e-4 of the total wind kinetic power being
converted into relativistic electron acceleration.
  Conclusions: This is the first time that the non-thermal emission from a CWB
is detected both in radio and high energies. This allows us to derive the most
robust constraints of the particle acceleration efficiency and magnetic field
intensity in a CWB so far, reducing the typical uncertainty of a few orders of
magnitude to just within a factor of two. This constitutes an important step
forward in our characterisation of the physical properties of CWBs.
