What does 'can't attack you' actually mean?
The other player can't attack anything you control (units or sites).
If some effect prevents me from giving the Peace Offering to the other player (i.e. King Arthur), can they attack me on their next turn?
No, if the other player does not end up controlling the Peace Offering at least for a moment, you have not successfully given it to them, and they will be free to attack you.
If Peace Offering leaves the realm, does the truce still apply?
Yes, once the Peace Offering has been given to your opponent, they can't attack you next turn, regardless of what happens to the Peace Offering after that.
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