  We consider energy sub-critical defocusing nonlinear wave equations on
$\mathbb{R}^3$ and establish the existence of unique global solutions almost
surely with respect to a unit-scale randomization of the initial data on
Euclidean space. In particular, we provide examples of initial data at
super-critical regularities which lead to unique global solutions. The proof is
based on probabilistic growth estimates for a new modified energy functional.
This work improves upon the authors' previous results in [25] by significantly
lowering the regularity threshold and strengthening the notion of uniqueness.
