Can there be interactions which destroy/create a single Fermion?

Consider a process of fundamental particles $frightarrow x,y$, where $x$ and $y$ are fundamental (no composites of fermions) bosonic particles.
Can such a process exist? I know that e.g. Majorana Fermions are not charged, but still they are Fermions and can only react to other Fermions.
An obstacle to such a process would be of course angular momentum, but can there be also half-integer angular momentum?
So could we write an interaction of the form $Deltamathcal{L} =gbar{psi}thetaPhi^2+ h.c.$, where $psi$ is a fermion $theta$ a Grassman variable and $Phi$ a scalar. Of course the matrix element would be Grassman-valued, but still the absolute value squared would still be a regular number if we choose a matrix representation for $theta$.
Could this be a valid interaction or are there other reasons why it must not exist (e.g. Pauli exclusion principle, unitarity, etc.)?