Fiber nonlinearities, that lead to nonlinear signal interference (NLI), are typically regarded as an instantaneous material response with respect to the optical field. However, in addition to an instantaneous part, the nonlinear fiber response consists of a delayed contribution, referred to as the Raman response. The imaginary part of its Fourier transform, referred to as the Raman gain spectrum, leads to inter-channel stimulated Raman scattering (ISRS). ISRS is a nonlinear effect that redistributes optical power from high to lower frequencies during propagation. However, as the nonlinear fiber response is causal, the Raman spectrum obeys the Kramers-Kronig relations resulting in the real part of the complex valued Raman spectrum. While the impact of the imaginary part (i.e. ISRS) is well studied, the direct implications of its associated real part on the NLI are unexplored. In this work, a theory is proposed to analytically quantify the impact of the real Raman spectrum on the nonlinear interference power. Starting from a generalized Manakov equation, an extension of the ISRS Gaussian Noise (GN) model is derived to include the real Raman spectrum and, thus, to account for the complete nonlinear Raman response. Accurate integral expressions are derived and approximations in closed-form are proposed. Different formulations for the case of single -and dual polarized signals are derived and novel analytical approximations of the real Raman spectrum are proposed. Moreover, it is analytically shown that the real Raman spectrum scales the strength of the instantaneous nonlinear distortions depending on the frequency separation of the interacting frequencies. A simple functional form is derived to assess the scaling of the NLI strength. The proposed theory is validated by numerical simulations over C-and C+L band, using experimentally measured fiber data.