  The formation and evolution of stellar clusters is intimately linked to that
of their host galaxies. To study this connection, we present the EMP-Pathfinder
suite of cosmological zoom-in Milky Way-mass simulations. These simulations
contain a sub-grid description for stellar cluster formation and evolution,
allowing us to study the simultaneous formation and evolution of stellar
clusters alongside their host galaxies across cosmic time. As a key ingredient
in these simulations, we include the physics of the multi-phase nature of the
interstellar medium (ISM), which enables studies of how the presence of a cold,
dense ISM affects cluster formation and evolution. We consider two different
star formation prescriptions: a constant star formation efficiency per
free-fall time, as well as an environmentally-dependent, turbulence-based
prescription. We identify two key results drawn from these simulations.
Firstly, we find that tidal shock-driven disruption caused by the graininess of
the cold ISM produces old ($\tau>10~$Gyr) stellar cluster populations with
properties that are in excellent agreement with the observed populations in the
Milky Way and M31. Importantly, the addition of the cold ISM addresses the
areas of disagreement found in previous simulations that lacked the cold gas
phase. Secondly, the formation of stellar clusters is extremely sensitive to
the baryonic physics that govern the properties of the cold, dense gas
reservoir in the galaxy. This implies that the demographics of stellar cluster
populations represent an important diagnostic tool for constraining baryonic
physics models in upcoming galaxy formation simulations that also include a
description of the cold ISM.
