  This paper is devoted to providing a unifying approach to the study of the
uniqueness of unconditional bases, up to equivalence and permutation, of
infinite direct sums of quasi-Banach spaces. Our new approach to this type of
problem permits us to show that a wide class of vector-valued sequence spaces
have a unique unconditional basis up to a permutation. In particular, solving a
problem from [F. Albiac and C. Ler\'anoz, Uniqueness of unconditional bases in
nonlocally convex $\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2,
394--401] we show that if $X$ is quasi-Banach space with a strongly absolute
unconditional basis then the infinite direct sum $\ell_{1}(X)$ has a unique
unconditional basis up to a permutation, even without knowing whether $X$ has a
unique unconditional basis or not. Applications to the uniqueness of
unconditional structure of infinite direct sums of non-locally convex Orlicz
and Lorentz sequence spaces, among other classical spaces, are also obtained as
a by-product of our work.
