The first supernovae enrich the previously pristine gas with metals, out of which the next generation of stars form. Based on hydrodynamical simulations, we develop a new stochastic model to predict the metallicity of star-forming gas in the first galaxies. On average, in internally enriched galaxies, the metals are well mixed with the pristine gas. However, in externally enriched galaxies, the metals can not easily penetrate into the dense gas, which yields a significant metallicity difference between the star-forming and average gas inside a halo. To study the consequences of this effect, we apply a semi-analytical model to Milky Way-like dark matter merger trees and follow stellar fossils from high redshift until the present day with a novel realistic metal mixing recipe. We calibrate the model to reproduce the metallicity distribution function (MDF) at low metallicities and find that a primordial IMF with slope of $mathrm{d}N/mathrm{d}M propto M^{-0.5}$ from $2 Msun$ to $180 Msun$ best reproduces the MDF. Our improved model for inhomogeneous mixing can have a large impact for individual minihalos, but does not significantly influence the modelled MDF at [Fe/H]$gtrsim -4$ or the best-fitting Pop~III IMF.
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