For the card Retreat (Awakenings #110), what happens if your opponent does not claim the field with that final action? Thanks in advance, Chris

Then I believe whoever has control of the battlefield will maintain control of the battlefield. If there is an action that occurs with claiming the battlefield, then that action does not occur if no one claims it. Just goes on and resets into the next action phase. Please correct if I'm wrong guys.

Long story short, claiming the battlefield and ending the round are entirely separate from each other. If no one claims the battlefield when the round ends (either through something like Retreat, or just both players passing in succession), then control of the battlefield won't change and the next round will start exactly as normal with the same player controlling the battlefield.