A facilitation tool that can help move groups to shared understanding involves everyone initially listing all the pros of an argument (written on a whiteboard or flipcharts for people to see), and when the entries are exhausted, then listing all the cons. Many times when the complete lists are placed side-by-side, people are able to form more objective conclusions and are able to become detached from the emotions surrounding the issues.
I successfully used this technique to reach a verdict with a jury that was near deadlock but once all the evidence in support of conviction was written out and all the rationale against it, enough people changed their vote to reach a majority decision. I’ve also seen its effectiveness with other groups when the discussion swirled around so many points that it was hard to weigh which argument was more compelling.
As a facilitator, you need to be vigilant to keep the listings singularly-focused (as there is great temptation to compile both lists simultaneously) but if successful you may find the solution that alluded you was there all along.