  The formation history of globular clusters (GCs) at redshift $z > 4$ remains
an unsolved problem. In this work, we use the cosmological, $N$-body
hydrodynamical ``zoom-in'' simulation GigaEris to study the properties and
formation of proto-GC candidates in the region surrounding the progenitor of a
Milky Way-sized galaxy. The simulation employs a modern implementation of
smoothed-particle hydrodynamics, including metal-line cooling and metal and
thermal diffusion and allows to resolve systems at the scale of star clusters.
We define proto-GC candidate systems as gravitationally bound stellar systems
with baryonic mass fraction $F_{\rm b} \geq 0.75$ and stellar velocity
dispersion $\sigma_{\star} < 20$ km s$^{-1}$. At $z=4.4$ we identify 9 systems
which satisfy our criteria, all of which form between 10 kpc to 30 kpc from the
centre of the main host. Their baryonic masses are in the range $10^5$- $10^7$
M$_{\odot}$. By the end of the simulation, they still have a relatively low
stellar mass ($M_{\star} \sim 10^4$--$10^5$ M$_{\odot}$) and a metallicity
($-1.8 \lesssim {\rm [Fe/H]} \lesssim -0.8$) similar to the blue Galactic GCs.
All of the identified systems except one appear to be associated with gas
filaments accreting onto the main galaxy in the circum-galactic region, and
formed at $z=5-4$. The exception is the oldest object, which appears to be a
stripped compact dwarf galaxy that has interacted with the main halo between $z
= 5.8$ and $z=5.2$ and has lost its entire dark matter content due to tidal
mass loss.
