In this paper, we construct the Rabinowitz-Floer homology for the coupled Dirac system \begin{equation*} \left\{ \begin{aligned} Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M,\\ Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M, \end{aligned} \right. \end{equation*} where M is an n-dimensional compact Riemannian spin manifold, D is the Dirac operator on M, and H#58;#92;SigmaM#92;oplus#92;SigmaM#92;to#92;mathbb#123;R#125; is a real valued superquadratic function of class C¹ with subcritical growth rates.
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