I too have discovered this major bug with ReLive. I am a big fan of AMD and really like the features included with ReLive, except the issue of not recording, streaming nor broadcasting audio when RealTek audio drivers are installed. My gaming monitor is on an HDMI link, so it uses AMD's audio driver, not Realtek's. That AMD or RealTek needs to do something about this, I agree.
If I could upgrade my audio, I would switch to a CreativeLabs SoundBlaster, but my HP Omen 17-an012dx is not upgradable in that way.