In my very limited understanding of geometric quantization, we quantize spin by choosing as our phase space $S^2$ with a suitably normalized area form as the symplectic form. Depending on the normalization we get out a finite-dimensional Hilbert space corresponding to some spin $j$ particle.
My question is if/how angular momentum addition may be understood in this picture. In the Hilbert-space picture, we can write things like $1/2\otimes 1/2=0\oplus 1$; i.e., we may decompose tensor products of reps of $SU(2)$ into irreps. If we naively try translating such an equation into phase space, I assume tensor products of Hilbert spaces correspond to Cartesian products of phase spaces, but I have no idea how direct sums would be interpreted (disjoint unions?). Does thinking carefully about quantizing products of spheres give a nice geometric interpretation of angular momentum addition in phase space?