  \emph{Ex ante} correlation is becoming the mainstream approach for
\emph{sequential adversarial team games}, where a team of players faces another
team in a zero-sum game. It is known that team members' asymmetric information
makes both equilibrium computation \textsf{APX}-hard and team's strategies not
directly representable on the game tree. This latter issue prevents the
adoption of successful tools for huge 2-player zero-sum games such as,
\emph{e.g.}, abstractions, no-regret learning, and subgame solving. This work
shows that we can recover from this weakness by bridging the gap between
sequential adversarial team games and 2-player games. In particular, we propose
a new, suitable game representation that we call
\emph{team-public-information}, in which a team is represented as a single
coordinator who only knows information common to the whole team and prescribes
to each member an action for any possible private state. The resulting
representation is highly \emph{explainable}, being a 2-player tree in which the
team's strategies are behavioral with a direct interpretation and more
expressive than the original extensive form when designing abstractions.
Furthermore, we prove payoff equivalence of our representation, and we provide
techniques that, starting directly from the extensive form, generate
dramatically more compact representations without information loss. Finally, we
experimentally evaluate our techniques when applied to a standard testbed,
comparing their performance with the current state of the art.
