According to quantum electrodynamics (QED), a strong electromagnetic field can make the vacuum state decay via the production of electron-positron pairs. Here we investigate the emission of soft photons which accompanies a nonperturbative process of pair production. Our analysis is carried out within the Furry picture to first order in the fine-structure constant. Also, it is shown that the presence of photons in the initial state gives rise to an additional (stimulated) channel of photon emission besides the pure vacuum one. On the other hand, the number of final (signal) photons includes also a negative contribution due to photon absorption within the pair production process. These contributions are evaluated and compared. To obtain quantitative predictions in the domain of realistic field parameters, we employ the Wentzel-Kramers-Brillouin approach. We propose using an optical-probe photon beam, whose intensity changes as it traverses a spatial region where a strong electric component of a background laser field is present. It is demonstrated that relative intensity changes on the level of 1% can be experimentally observed once the intensity of the strong background field exceeds 1027 W/cm2 within a large laser-wavelength interval. This finding is expected to significantly support possible experimental investigations of nonlinear QED phenomena in the nonperturbative regime.