  Context. Recent astrometric and spectroscopic surveys of OB stars have
revealed a few stellar-mass black holes (BHs) with orbital periods as low as 10
days. No X-ray counterpart has been detected, due to the absence of a
radiatively efficient accretion disk around the BH. Yet, dissipative processes
in the hot, dilute and strongly magnetized plasma around the BH (so-called BH
corona) can still lead to non-thermal X-ray emission (e.g. synchrotron).
  Aims. We determine the X-ray luminosity distribution from BH+OB star binaries
up to orbital periods of a few thousand days.
  Methods. We use detailed binary evolution models computed with MESA for
initial primary masses of 10-90 $M_{\odot}$ and orbital periods from 1-3000 d.
The X-ray luminosity is computed for a broad range of radiative efficiencies.
  Results. We show that particle acceleration through magnetic reconnection can
heat the BH corona. A substantial fraction of the gravitational potential
energy from the accreted plasma is converted into non-thermal X-ray emission.
Our population synthesis analysis predicts at least 28 (up to 72) BH+OB star
binaries in the Large Magellanic Cloud (LMC) to produce X-ray luminosity above
10$^{31}$ erg$\,$s$^{-1}$, observable through focused Chandra observations. We
identify a population of SB1 systems in the LMC and HD96670 in the Milky Way
comprising O stars with unseen companions of masses above 2.3 $M_{\odot}$ that
aligns well with our predictions. The predicted luminosities of the OB
companions to these X-ray-emitting BHs are 10$^{4.5-5.5}$ $L_{\odot}$.
  Conclusions. These results make the case for long-time exposure in X-rays of
the stellar-mass BH candidates identified around OB stars. It will constrain
the underlying population of X-ray-faint BHs, the evolution from single to
double degenerate binaries, and the progenitors of gravitational wave mergers.
(Abridged)
