Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in P^8 as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU_C(2, L), the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of G(2, 8). In fact, each point p ∈ C defines a natural embedding of SU_C(2, O(p)) in G(2, 8). We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU_C(2, O(p)), and thus deserves to be coined the Coble quadric of the pointed curve (C, p).