  Context. The star-black hole (S-BH) binary known as Gaia BH3, discovered by
the Gaia Collaboration is chemically and kinematically associated with the
metal-poor ED-2 stream in the Milky Way halo.
  Aims. We explore the possibility that Gaia BH3 was assembled dynamically in
the progenitor globular cluster (GC) of the ED-2 stream.
  Methods. We used a public suite of star-by-star dynamical Monte Carlo models
to identify S-BH binaries in GCs with different initial masses and (half-mass)
radii.
  Results. We show that a likely progenitor of the ED-2 stream was a relatively
low-mass ($\lesssim10^5M_\odot$) GC with an initial half-mass radius of ~4 pc.
Such a GC can dynamically retain a large fraction of its BH population and
dissolve on the orbit of ED-2. From the suite of models we find that GCs
produce ~ 3 - 30 S-BH binaries, approximately independent of initial GC mass
and inversely correlated with initial cluster radius. Scaling the results to
the Milky Way GC population, we find that ~75% of the S-BH binaries formed in
GCs are ejected from their host GC, all in the early phases of evolution
($\lesssim1$ Gyr); these are expected to no longer be close to streams. The
~25% of S-BH binaries retained until dissolution are expected to form part of
streams, such that for an initial mass of the progenitor of ED-2 of a few
$10^4M_\odot$, we expect ~2-3 S-BH to end up in the stream. GC models with
metallicities similar to Gaia BH3 ($\lesssim1\%$ solar) include S-BH binaries
with similar BH masses ($\gtrsim30M_\odot$), orbital periods, and
eccentricities.
  Conclusions. We predict the Galactic halo contains of order $10^5$ S-BH
binaries that formed dynamically in GCs, a fraction of which may readily be
detected in Gaia DR4. The detection of these sources provides valuable tests of
BH dynamics in clusters and the possible role in formation of gravitational
wave sources.
