Auger-like exciton-exciton annihilation (EEA) is considered the key fundamental limitation to quantum yield in devices based on excitons in two-dimensional (2d) materials. Since it is challenging to experimentally disentangle EEA from competing processes, guidance of a quantitative theory is highly desirable. The very nature of EEA requires a material-realistic description that is not available to date. We present a many-body theory of EEA based on first-principle band structures and Coulomb interaction matrix elements that goes beyond an effective bosonic picture. Applying our theory to monolayer MoS$_2$ encapsulated in hexagonal BN, we obtain an EEA coefficient in the order of $10^{-3}$ cm$^{2}$s$^{-1}$ at room temperature, suggesting that exciton annihilation is often dominated by other processes, such as defect-assisted scattering. Our studies open a perspective to quantify the efficiency of intrinsic EEA processes in various 2d materials in the focus of modern materials research.
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