We describe the electron energy spectra and half-lives of forbidden $\beta$ decays for nuclear systems outside the $^{78}$Ni core in the framework of the realistic shell model, starting from a realistic interaction. The effective shell-model Hamiltonians and
decay operators are derived using many-body perturbation theory. In particular, we examine how sensitive the calculated electron energy spectra and half-lives are to the renormalization of forbidden $\beta$-decay operators. The study focuses on the first-forbidden decays of $^{89}$Sr and $^{90}$Y , on the second-forbidden decays of $^{94}$Nb and $^{99}$Tc, as well as the fourth-forbidden decays of $^{113}$Cd and $^{115}$In.