  In most Dirac semimetals, time-reversal and inversion symmetries are believed
to play a crucial role in their stability. We demonstrate that these symmetries
are broken in Dirac fermions in the organic conductor
$\alpha$-(BEDT-TTF)$_2$I$_3$ due to the strong electronic correlation. The
system is a three-dimensional type-II Dirac semimetal in the coherent
inter-layer tunneling regime. A chiral anomaly is predicted to be observed in
the magnetoresistance when the magnetic field is tuned to the inter-layer
tunneling direction. Our result suggests that $\alpha$-(BEDT-TTF)$_2$I$_3$ is a
useful platform to explore the interplay between the chiral anomaly and the
strong correlation and/or dimensionality.
