Arkham Horror
Does Tattered Cloak stop monsters from engaging me, even a Massive one after I evade it?
Yes, for monsters that are not Epic. While you wear it, those monsters do not engage you, including a Massive monster that re-engages after you evade it. Attacking an engaged monster suspends the card's effect until that monster is no longer engaged, and the rules give no way to switch the item off on purpose.