  We show that the imaginary multiplicative chaos $\exp(i\beta \Gamma)$
determines the gradient of the underlying field $\Gamma$ for all log-correlated
Gaussian fields with covariance of the form $-\log |x-y| + g(x,y)$ with mild
regularity conditions on $g$, for all $d \geq 2$ and for all $\beta \in
(0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary
Gaussian free field is measurable w.r.t. its imaginary chaos.
