I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:

(1) The spinor bundle $S = S_+ \oplus S_-$ associated to a spin or spinc structure of Riemannian four-manifold $M$

(2) If the manifold $M$ has almost complex structure, then the bundle $\tilde S \equiv \Lambda^{0,*}T^*M$ is also a spin bundle, which can also be split into $\tilde S_+ = \Lambda^{0,0}T^*M \oplus \Lambda^{0,2}T^*M$ and $\tilde S_- = \Lambda^{0,1}T^*M$.

My question is: are the two examples the same? In particular, I know that $S_+ \otimes S_+ = \Lambda_+^2 T^*M \oplus C^\infty(M)$ where $\Lambda_+^2T^*M$ denotes the bundle of self-dual two forms (corresponding to symmetric product of two $S_+$), I wonder if the same is true for $\tilde S_+$? I think it is true, since I can multiply by $(\sigma_{\mu \nu})^{\alpha \beta}$ to go from a bi-spinor to a self-dual 2-form. But I want to see if there is a more geometrical way (playing with the $(0,{\rm even})$-forms) to see it.

Thanks!