Throughout, I think of games and their underlying trees as the same: so a "clopen game" and a "well-founded tree" mean the same thing.

Fix a sequence of clopen games $\lbrace T_i: i\in\omega\rbrace$. We can form the open game $\overline{(T_i)}$, which is played as follows. First, players I and II play the game $T_0$; then if player I wins, they play the game $T_1$; then if player I wins, they play the game $T_2$; and so on, and if player II ever wins one of the $T_i$ then player II wins the whole of $\overline{(T)_i}$, and otherwise I wins $\overline{(T_i)}$. Note that the sequence of games played is fixed, and does not depend on how the players play within any specific game, so this is a very deterministic way of pasting games together.

Essentially, $\overline{(T_i)}$ is gotten by pasting together the clopen games $T_i$. My question is the following. It's easy to show that not every open game $T$ is of the form $\overline{(T_i)}$ for some sequence of clopen games $T_i$. Is there a simple description of the open games which are of this form? If so, what are such games called?