  We are concerned with the problem of global well-posedness of the 3D
Navier--Stokes equations on the torus with unitary viscosity. While a full
answer to this question seems to be out of reach of the current techniques, we
establish a regularization by a deterministic vector field. More precisely, we
consider the vorticity form of the system perturbed by an additional transport
type term. Such a perturbation conserves the enstrophy and therefore a priori
it does not imply any smoothing. Our main result is a construction of a
deterministic vector field $v=v(t,x)$ which provides the desired regularization
of the system and yields global well-posedness for large initial data outside
arbitrary small sets. The proof relies on probabilistic arguments developed by
Flandoli and Luo, tools from rough path theory by Hofmanov\'a, Leahy and
Nilssen and a new Wong--Zakai approximation result, which itself combines
probabilistic and rough path techniques.
