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