  Millisecond pulsars are very likely the main source of gamma-ray emission
from globular clusters. However, the relative contributions of two separate
emission processes--curvature radiation from millisecond pulsar magnetospheres
vs. inverse Compton emission from relativistic pairs launched into the globular
cluster environment by millisecond pulsars--have long been unclear. To address
this, we search for evidence of inverse Compton emission in 8-year
$\textit{Fermi}$-LAT data from the directions of 157 Milky Way globular
clusters. We find a mildly statistically significant (3.8$\sigma$) correlation
between the measured globular cluster gamma-ray luminosities and their photon
field energy densities. However, this may also be explained by a hidden
correlation between the photon field densities and the stellar encounter rates
of globular clusters. Analysed $\textit{in toto}$, we demonstrate that the
gamma-ray emission of globular clusters can be resolved spectrally into two
components: i) an exponentially cut-off power law and ii) a pure power law. The
latter component--which we uncover at a significance of 8.2$\sigma$--has a
power index of 2.79 $\pm$ 0.25. It is most naturally interpreted as inverse
Compton emission by cosmic-ray electrons and positrons injected by millisecond
pulsars. We find the luminosity of this power-law component is comparable to,
or slightly smaller than, the luminosity of the curved component, suggesting
the fraction of millisecond pulsar spin-down luminosity into relativistic
leptons is similar to the fraction of the spin-down luminosity into prompt
magnetospheric radiation.
