This report summarizes the work of the EMMI Rapid Reaction Task Force on "Real and Virtual Photon Production at Ultra-Low Transverse Momentum and Low Mass at the LHC". We provide an overview of the soft-photon puzzle, i.e., of the long-standing discrepancy between experimental data and predictions based on Low's soft-photon theorem, also referred to as "anomalous" soft photon production, and we review the current theoretical understanding of soft radiation and soft theorems. We also focus on low-mass dileptons as a tool for determining the electrical conductivity of the medium produced in high-energy nucleus-nucleus collisions. We discuss how both topics can be addressed with the planned ALICE 3 detector at the LHC.