While the discussion has mainly focused on the battlefield priests and wether or not they are suspect from a realism aspect.
But what hasn't been looked at is what priests DO. I'm sorry, but I find a little hard to believe that troops engaged in a battle would notice a group of dress wearing men waving crosses, let alone take heart from tem. They noise of battle would prevent it. This IMO strengthens the case for making priests retinue only because realisticly your only chance to show your men that God is on your side is prior to battle. Once battle is joined priest would find it impossible to make any impact what-so-ever. What are the priests going to do? Tap a soldier on the shoulder mid battle, asking him about God? I don't thinks so.
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