  We study the incompressible hypodissipative Navier--Stokes equations with
dissipation exponent $0 < \alpha < \frac{1}{2}$ on the three-dimensional torus
perturbed by an additive Wiener noise term and prove the existence of an
initial condition for which distinct probabilistic weak solutions exist. To
this end, we employ convex integration methods to construct a pathwise
probabilistically strong solution, which violates a pathwise energy inequality
up to a suitable stopping time. This paper seems to be the first in which such
solutions are constructed via Beltrami waves instead of intermittent jets or
flows in a stochastic setting.
