A generalization of the band method is presented. In the new set up there are two semi-band structures, a right one and a left one, and the two are coupled. Moreover the role of the band extension in the classical version is taken over by a special positive real part extension. It is shown that with minor modifications the main results of the band method extend to this more general set up. As an application a generalization of the Carathéodory-Toeplitz interpolation problem for operator functions with operator arguments is solved.