  The initial value problem for the general coupled Hirota system with nonzero
boundary conditions at infinity is solved by reporting a rigorous theory of the
inverse scattering transform. With the help of a suitable uniformization
variable, both the inverse and the direct problems are analyzed which allows us
to develop the inverse scattering transform on the complex $z$-plane. Firstly,
analyticity of the scattering eigenfunctions and scattering data, properties of
the discrete spectrum, symmetries, and asymptotics are discussed in detail.
Moreover, the inverse problem is posed as a Riemann-Hilbert problem for the
eigenfunctions, and the reconstruction formula of the potential in terms of
eigenfunctions and scattering data is presented. Finally, the main
characteristics of these obtained soliton solutions are graphically discussed
in the $2\times2$ self-focusing case. This family of solutions contains novel
Akhmediev breather and Kuznetsov-Ma soliton. These results would be of much
importance in understanding and enriching breather wave phenomena arising in
nonlinear and complex systems, especially in Bose-Einstein condensates.
