  We construct a bosonic analog of a two-dimensional topological Dirac
Semi-Metal (DSM). The low-energy description of the most basic 2D DSM model
consists of two Dirac cones at positions $\pm\mathbf{k}_0$ in momentum space.
The local stability of the Dirac cones is guaranteed by a composite symmetry
$Z_2^{\mathcal{TI}}$, where $\mathcal{T}$ is time-reversal and $\mathcal{I}$ is
inversion. This model also exhibits interesting time-reversal and inversion
symmetry breaking electromagnetic responses. In this work we construct a
bosonic version by replacing each Dirac cone with a copy of the $O(4)$
Nonlinear Sigma Model (NLSM) with topological theta term and theta angle
$\theta=\pm \pi$. One copy of this NLSM also describes the gapless surface
termination of the 3D Bosonic Topological Insulator (BTI). We compute the
time-reversal and inversion symmetry breaking electromagnetic responses for our
model and show that they are twice the value one gets in the DSM case matching
what one might expect from, for example, a bosonic Chern insulator. We also
investigate the stability of the BSM model and find that the composite
$Z_2^{\mathcal{TI}}$ symmetry again plays an important role. Along the way we
clarify many aspects of the surface theory of the BTI including the
electromagnetic response, the charges and statistics of vortex excitations, and
the stability to symmetry-allowed perturbations. We briefly comment on the
relation between the various descriptions of the $O(4)$ NLSM with $\theta=\pi$
used in this paper (a dual vortex description and a description in terms of
four massless fermions) and the recently proposed dual description of the BTI
surface in terms of $2+1$ dimensional Quantum Electrodynamics with two flavors
of fermion ($N=2$ QED$_3$). In a set of four Appendixes we review some of the
tools used in the paper, and also derive some of the more technical results.
