Building on ideas of Shelah’s Maximal failure of sequence locality in AECs, we give converses to results of Tameness from Large Cardinal Axioms by adding “almost” to the large cardinal properties (the importance of , e. g., almost strongly compact cardinals was also discovered by Brooke-Taylor and Rosicky in Accessible Images Revisited). This (and the work of Brooke-Taylor and Rosicky) allow us to prove a triple equivalence between a) the existence of class many strongly compact cardinals, b) the statement that all AECs are eventually tame, and c) a category-theoretic principle originally isolated by Makkai and Pare.