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Abstract In this paper, we develop the Riemann–Hilbert approach to the inverse scattering transform (IST) for the complex coupled short‐pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real short‐pulse equation (SPE) and complex short‐pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one‐soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather.