  In this paper we introduce and study a Bargmann-Radon transform on the real
monogenic Bargmann module. This transform is defined as the projection of the
real Bargmann module on the closed submodule of monogenic functions spanned by
the monogenic plane waves. We prove that this projection can be written in
integral form in terms the so-called Bargmann-Radon kernel. Moreover, we have a
characterization formula for the Bargmann-Radon transform of a function in the
real Bargmann module in terms of its complex extension and then its restriction
to the nullcone in $\mathbb C^m$. We also show that the formula holds for the
Szeg\H{o}-Radon transform that we introduced in [4] (see list of references).
Finally, we define the dual transform and we provide an inversion formula.
