So your group needs to make a decision and here they are at the same impasse that has derailed the process before. Time to do something different.
Three strategies you can employ right now:
1. Change the context to get everyone out of a predictable and often comfortable business-as-usual setting. This could be simple – like changing the environment from boardroom to, say, a city street, or more complex: crafting the elements of the issue into a game structure, or perhaps a metaphor of some kind; Ex: “We’ve been approaching this issue like it’s a broken down bus in need of mechanical parts, but maybe we can look at it like an untrained race horse, in need of a clear training regimen”.
2. Shift the emotional mood: Groups can be stuck in multiple emotional states and not all of them are angry or hostile. Some groups get along perfectly well socially, and nothing at all gets done. Other groups are distracted, fractured, unfocused. What can you do to shift the general mood of your group into a more productive mind space? Hold the meeting standing up or walking to counter the “lazyboy” effect? Start with a humorous activity to cut through the tension?
3. Make sure everyone is listening and really being heard. This is not as easy as it might sound, but the juncture at which people stop listening is the exact point where positions become hardened and movement is exponentially more difficult. You can slow the discussion down, get people to repeat what they just heard, and ask participants to clarify or re-phrase anything misunderstood. Also, you can highlight the few points people actually agree on, as well as what accomplishments, if any, the group has already managed to achieve. Much of what we consider hostility in a group is fueled by an inability to feel authentically heard and understood. If this is addressed satisfactorily, along with the other suggestions, your group will be on its way to effective progress.