Remove or Reduce the Perceived Threat
A key cause of anger or conflict is that people may perceive that they, or things they hold dear, are threatened. Perhaps they feel that something being discussed threatens their reputation, judgment, chances of leading a successful project, or chances of getting a bonus. Or perhaps they perceive a threat to a project they've worked hard to promote, or believe in strongly.
There are two parts to this: the perception of threat, and the threat itself.
This is where you need to explore the issue and fully understand what it is. It's possible that the perception may be wrong – perhaps based on faulty or incomplete information. Here you need to supply the correct information. Or it may be that the perception is correct, and the person is right to feel threatened. Here you need to address the situation.
Another thing you can do is make sure that you clear up unknowns, because the unknown is often treated as a threat. Going back again to our example of the Alison vs James hiring decision, you might ask the supporters of each to talk about what benefits their non-preferred candidate would bring to the team, and what areas for development they'd need to work on.