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Our goal in this paper is to advance the state of the art of the topic ofuniqueness of unconditional basis. To that end we establish general conditionson a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space$\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that everyunconditional basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splitsinto unconditional bases of each summand. As application of our methods weobtain that, among others, the spaces $H_p(\mathbb{T}^d)\oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$and $d\in\mathbb{N}$, have a unique unconditional basis (up to equivalence andpermutation).

Uniqueness of unconditional basis of $H_p(\mathbb{T})\oplus\ell_{2}$ and $H_p(\mathbb{T})\oplus\mathcal{T}^{(2)}$ for $0