Detecting the thermal and non-thermal emission from the shocked cosmic gas surrounding large-scale structures represents a challenge for observations, as well as a unique window into the physics of the warm-hot intergalactic medium. In this work, we present synthetic radio and X-ray surveys of large cosmological simulations in order to assess the chances of jointly detecting the cosmic web in both frequency ranges. We then propose best observing strategies tailored for existing (LOFAR, MWA, and XMM) or future instruments (SKA-LOW and SKA-MID, Athena, and eROSITA). We find that the most promising targets are the extreme peripheries of galaxy clusters in an early merging stage, where the merger causes the fast compression of warm-hot gas onto the virial region. By taking advantage of a detection in the radio band, future deep X-ray observations will probe this gas in emission, and help us to study plasma conditions in the dynamic warm-hot intergalactic medium with unprecedented detail.