Suppose $(X_t,\mathcal F_t)_{t \in \mathbb Z}$ is such that $\mathbb E(X_t\mid \mathcal F_{t-1}) = 0$ for every $t$, where $(\mathcal F_t)$ is a filtration on a probability space $(\Omega,\mathcal F,\mathbb P)$. But $(X_t)$ is not necessarily adapted to $(\mathcal F_t)$.
Then is there any well-known situation or fact by which we can ensure the existence of some filtration $(\mathcal G_t)$ with respect to which $(X_t)$ is adapted and still $\mathbb E(X_t\mid \mathcal G_{t-1}) = 0$ for every $t$?
That is, is there any special example or condition under which we can make a sequence or an array of random variables a martingale difference by changing(enlarging) the original filtration? My naive guess is that one could realize this by using the natural filtration of $(X_t)$ and the original filtration $(\mathcal F_t)$