We study the existence of local and global strong solutions for the stochastic tamed Navier--Stokes equations on the whole space $\mathbb{R}^3$, driven by multiplicative Wiener noise and compensated Lévy jump noise. For $p > 3$, we first prove the existence of a pathwise unique maximal local $L^p$-strong solution for divergence-free, $\mathcal{F}_0$-measurable initial data in $L^p(Ω; L^p(\mathbb{R}^3;\mathbb{R}^3))$. For initial data additionally belonging to $L^2(Ω; H^1(\mathbb{R}^3;\mathbb{R}^3))$, we overcome the non-local pressure obstruction inherent to the whole space, to establish the existence of a pathwise unique global strong solution.