Angular Momentum Addition in Phase Space QM 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?
 A: I could give you an answer by barking up a very different tree indeed! In phase space QM, and not, repeat not geometric quantization, you may work on flat phase spaces and forfeit spheres altogether, the way you actually do in Hilbert spaces. If you can stomach that, read on, otherwise not, lest you feel your expectations betrayed. 
Plain vanilla 
phase-space QM in a 2d phase space 
replicates the standard ladder algebraic features of quantized angular momentum in Hilbert space, with very parallel algebraic tricks, cf Exercise 0.8 of p.33 of that book.  A long moment's thought should convince you then that Hilbert space addition of angular momentum maps isomorphically to a  phase space equivalent, on tensor-product phase spaces, one-to-one corresponding to the Hilbert spaces. The Hilbert space comultiplication representing Kronecker multiplication of SU(2),
$$
\Delta(J_a)\equiv J_a \otimes 1\!\!\!1 +  1\!\!\!1 \otimes J_a
$$
will satisfy the same Lie Algebra as the left-and-right $J_a$s, and the Casimir $\Delta(J_a)\Delta(J_a)$ will be reducible to a direct sum of the Casimirs of the Clebsched reduced reps, so, then the respective eigenvalues times the respective identities in the block subspaces of each irrep.  
The linear algebra is identical in phase space, but, as you might recall, operator multiplication amounts to star-multiplication of plain phase-space functions in each phase space of the tensor product. To eschew spinors as a first step, try Kronecker-multiplying two spin 1s, so, then  3⊗3=5⊕3⊕1, so, then, spin 2, 1, and 0. Half-integral spins can then be introduced in standard ways, cf. Brif & Mann. 
