In the QCD+QED plasma, the transport properties of electromagnetic (EM) probes such as leptons and photons could be modified by the lepton/photon-parton scattering compared with the case in the QED plasma. We investigate the shear viscosity of thermalized leptons and photons depending on the coupling of the QCD plasma, which is in practice approximated by the N=4 super Yang-Mills (SYM) gauge theory at finite temperature. It is found that the lepton shear viscosity due to the lepton-quark scattering is inversely proportional to the ratio of electric conductivity of the QCD plasma to temperature up to the leading logarithmic order of the EM coupling and is suppressed compared with the one from lepton-lepton scattering. On the other hand, the photon shear viscosity up to the leading order of the EM coupling is suppressed by the photon-parton scattering, where the suppression is favored by the strong coupling of the QCD plasma. On the contrary, the lepton shear viscosity behaves oppositely due to the decrease of electric conductivity of the QCD plasma at stronger coupling. That is, the leptons become more viscous but photons become more fluid-like in the strongly coupled QCD plasma. We suggest that the collective behavior of thermalized photons may generate anisotropic flow in the late-time evolution of an expanding system.