  We study the parabolic defocusing stochastic quantization equation with both
mutliplicative spatial white noise and an independant space-time white noise
forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing
the nonlinearity, we construct the random Gibbs measure as an absolutely
continuous measure with respect to the law of the Anderson Gaussian Free Field
for fixed realization of the spatial white noise. Then, when the initial data
is distributed according to the Gibbs measure, we prove almost sure global
well-posedness for the dynamics and invariance of the Gibbs measure.
