We present 25 cosmological zoom-in simulations of Milky Way-mass galaxies in the `MOdelling Star cluster population Assembly In Cosmological Simulations within EAGLE (E-MOSAICS) project. E-MOSAICS couples a detailed physical model for the formation, evolution, and disruption of star clusters to the EAGLE galaxy formation simulations. This enables following the co-formation and co-evolution of galaxies and their star cluster populations, thus realising the long-standing promise of using globular clusters (GCs) as tracers of galaxy formation and assembly. The simulations show that the age-metallicity distributions of GC populations exhibit strong galaxy-to-galaxy variations, resulting from differences in their evolutionary histories. We develop a formalism for systematically constraining the assembly histories of galaxies using GC age-metallicity distributions. These distributions are characterised through 13 metrics that we correlate with 30 quantities describing galaxy formation and assembly (e.g. halo properties, formation/assembly redshifts, stellar mass assembly time-scales, galaxy merger statistics), resulting in 20 statistically (highly) significant correlations. The GC age-metallicity distribution is a sensitive probe of the mass growth, metal enrichment, and minor merger history of the host galaxy. No such relation is found between GCs and major mergers, which play a sub-dominant role in GC formation for Milky Way-mass galaxies. Finally, we show how the GC age-metallicity distribution enables the reconstruction of the host galaxys merger tree, allowing us to identify all progenitors with masses $M_*gtrsim10^8$ M$_odot$ for redshifts $1leq zleq2.5$. These results demonstrate that cosmological simulations of the co-formation and co-evolution of GCs and their host galaxies successfully unlock the potential of GCs as quantitative tracers of galaxy formation and assembly.
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