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Using F-theory we construct 4D ${\cal N}=1$ SUGRA theories with the StandardModel gauge group, three chiral generations, and matter parity in order toforbid all dimension four baryon and lepton number violating operators. Theunderlying geometries are derived by constructing smooth genus-one fiberedCalabi--Yau fourfolds using toric tops that have a Jacobian fibration with rankone Mordell--Weil group and $SU(3) \times SU(2) $ singularities. The necessarygauge backgrounds on the smooth fourfolds are shown to be fully compatible withthe quantization condition, including positive integer D3-tadpoles. Thisconstruction realizes for the first time a consistent UV completion of anMSSM-like model with matter parity in F-theory. Moreover our construction isgeneral enough to also exhibit other relevant $\mathbb{Z}_2$ charge extensionsof the MSSM such as lepton and baryon parity. Such models however are renderedinconsistent by non-integer fluxes, which are necessary for producing the exactMSSM chiral spectrum. These inconsistencies turn out to be intimately relatedto field theory considerations regarding a UV-embedding of the $\mathbb{Z}_2$into a $U(1)$ and the resulting discrete anomalies.

An F-theory Realization of the Chiral MSSM with $\mathbb{Z}_2$-Parity