If I look on D&D5 rules (
here in french), they should have at least :
- a different outfit (equivalent to Shadowheart chainmail and/or an armor like the flaming fist with a tabard which takes the symbol and the colors of the god/goddess)
- access to a limited number of spells defined by the deity's domain and level
- access to some special capacities according to the deity's domain and level
Specific interactions between clerics is cool but difficult to implement.
Limitations of powers due to actions not aligned also but lead to an approval system of the actions of the player. In multiplayer, it can become tricky.
