  We establish pathwise existence of solutions for porous media and fast
diffusion equations with nonlinear gradient noise, in the full regime
$m\in(0,\infty)$ and for any initial data in $L^2$. Moreover, if the initial
data is positive, solutions are pathwise unique. In turn, the solution map of
these equations is a continuous function of the driving noise and it generates
an associated random dynamical system. Finally, in the regime
$m\in\{1\}\cup(2,\infty)$, all the aforementioned results also hold for signed
initial data.
