  The ultra-diffuse galaxy (UDG) NGC1052-DF2 has a low dark matter content and
hosts a very unusual globular cluster (GC) population, with a median luminosity
$\sim4$ times higher than in most galaxies and containing about 5~per~cent of
the galaxy's stars. We apply a theoretical model that predicts the initial
cluster mass function as a function of the galactic environment to investigate
the origin of DF2's peculiar GC system. Using the GC mass function, the model
constrains the star-forming conditions in the galaxy during the formation of
its GCs, $\sim9~\rm{Gyr}$ ago. We predict that the GCs formed in an environment
with very high gas surface density, $\Sigma_{\rm ISM} \gtrsim 10^3 M_{\odot}
\rm{pc}^{-2}$, and strong centrifugal support, $\Omega \gtrsim
0.7~\rm{Myr}^{-1}$, similar to nearby circum-nuclear starbursts and the central
region of the Milky Way. These extreme conditions required to form the observed
GC population imply a very high cluster formation efficiency of $\gtrsim 78$
per cent, and contrast strongly with the current diffuse nature of the galaxy.
Since a nuclear starburst would lead to the rapid in-spiral of the GCs and is
ruled out by the absence of a nuclear star cluster, we propose that the GCs
plausibly formed during a major merger at $z\sim1.3$. The merger remnant must
have undergone significant expansion of its stellar (and perhaps also its dark
matter) component to reach its low present surface brightness, leading to the
interesting possibility that it was the formation of DF2's extreme GC
population that caused it to become a UDG. If true, this strong structural
evolution would have important implications for understanding the origins of
UDGs.
