  We give a condition which characterises those weight structures on a derived
category which come from a Thomason filtration on the underlying scheme. Weight
structures satisfying our condition will be called $\otimes ^c$-weight
structures. More precisely, for a Noetherian separated scheme $X$, we give a
bijection between the set of compactly generated $\otimes ^c$-weight structures
on $\mathbf{D} (\mathrm{Qcoh\hspace{1mm}}X)$ and the set of Thomason
filtrations of $X$. We achieve this classification in two steps. First, we show
that the bijection of S\v{t}ov\'{\i}\v{c}ek and Posp\'{\i}\v{s}il restricts to
give a bijection between the set of compactly generated $\otimes ^c$-weight
structures and the set of compactly generated tensor t-structures. We then use
our earlier classification of compactly generated tensor t-structures to obtain
the desired result. We also study some immediate consequences of these
classifications in the particular case of the projective line. We show that in
contrast to the case of tensor t-structures, there are no non-trivial tensor
weight structures on $\mathbf{D}^b (\mathrm{Coh \hspace{1mm}} \mathbb{P}^1_k)$.
