  We introduce and develop a set-based semantics for asynchronous TeamLTL. We
consider two canonical logics in this setting: the extensions of TeamLTL by the
Boolean disjunction and by the Boolean negation. We establish fascinating
connections between the original semantics based on multisets and the new
set-based semantics as well as show one of the first positive complexity
theoretic results in the temporal team semantics setting. In particular we show
that both logics enjoy normal forms that can be utilised to obtain results
related to expressivity and complexity (decidability) of the new logics. We
also relate and apply our results to recently defined logics whose
asynchronicity is formalized via time evaluation functions.
