  The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian
reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and
their simple quotients have a long history of applications in conformal field
theory and string theory. Their representation theories are therefore quite
interesting. Here, we classify the simple relaxed highest-weight modules, with
finite-dimensional weight spaces, for all admissible but nonintegral levels,
significantly generalising the known highest-weight classifications
[arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple
Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are
always rational in category $\mathscr{O}$, whilst they always admit
nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in
\mathbb{Z}_{\ge0}$.
