  We construct H\"older continuous, global-in-time probabilistically strong
solutions to 3D Euler equations perturbed by Stratonovich transport noise.
Kinetic energy of the solutions can be prescribed a priori up to a stopping
time, that can be chosen arbitrarily large with high probability. We also prove
that there exist infinitely many H\"older continuous initial conditions leading
to non-uniqueness of solutions to the Cauchy problem associated with the
system. Our construction relies on a flow transformation reducing the SPDE
under investigation to a random PDE, and convex integration techniques
introduced in the deterministic setting by De Lellis and Sz\'ekelyhidi, here
adapted to consider the stochastic case. In particular, our novel approach
allows to construct probabilistically strong solutions on $[0,\infty)$
directly.
