I've read the following result without proof from my lecture notes:
Let $X$ and $Y$ be two continuous local martingales (on the same probability space) with reducing sequences and filtrations $(\mathcal{T}_n)_{n\in \mathbb{N}},(\mathscr{F}_t)_{t\geq 0}$ and $(\mathcal{S}_n)_{n\in \mathbb{N}},(\mathscr{G_t})_{t\geq 0}$ respectively then there exists an $a.s.$ unique finite variation process started at zero, $\langle X, Y\rangle$, such that
$$XY -\langle X, Y\rangle$$
is a continuous local martingale.
My question is with respect to which filtration and which reducing sequence is this new process a local martingale?
It's clear to me that if the local martingales are independent, then we can define a new filtration $\mathscr{H}_t=\sigma(\mathscr{F}_t, \mathscr{G}_t)$ and there are no problems, so I'm interested in the case when they are not independent.