A growing body of work has shown that AI-assisted methods -- leveraging large language models, social choice methods, and collective dialogues -- can help navigate polarization and surface common ground in controlled lab settings. But what can these approaches contribute in real-world contexts? We present a case study applying these techniques to find common ground between Israeli and Palestinian peacebuilders in the period following October 7th, 2023. From April to July 2024 an iterative deliberative process combining LLMs, bridging-based ranking, and collective dialogues was conducted in partnership with the Alliance for Middle East Peace. Around 138 civil society peacebuilders participated including Israeli Jews, Palestinian citizens of Israel, and Palestinians from the West Bank and Gaza. The process resulted in a set of collective statements, including demands to world leaders, with at least 84% agreement from participants on each side. In this paper, we document the process, results, challenges, and important open questions.