  We show global existence and non-uniqueness of probabilistically strong,
analytically weak solutions of the three-dimensional Navier-Stokes equations
perturbed by Stratonovich transport noise. We can prescribe either: \emph{i})
any divergence-free, square integrable intial condition; or \emph{ii}) the
kinetic energy of solutions up to a stopping time, which can be chosen
arbitrarily large with high probability. Solutions enjoy some Sobolev
regularity in space but are not Leray-Hopf.
