The games I've come across thus far all have the property that their principal ideals are finite i.e the game terminates after finitely many moves so that the last position has the form {A|B} with (at least one of) A or B being empty.
How would one define "winning" in a game whose game tree has either an infinite principal ideal or, more generally, an infinite chain?
There are two common approaches in the sphere of Combinatorial Game Theory.
One approach is to declare that if a game goes on forever, it is a draw. Instead of four outcomes like "positive, negative, zero, or fuzzy", there are now nine: If Left goes first, they can either guarantee a win in finite time, or Right can, or both can at best guarantee a draw. Similarly if Right moves first, for a total of $3*3=9$ outcomes. An online resource discussing the main points in this theory of "loopy" games is Coping with cycles by Aaron N. Siegel. Most (all?) of that material and more can be found in Chapter VI of Siegel's Combinatorial Game Theory.
Another approach is to grant infinite play as a win to one player. This is done in John H. Conway's Angel Problem, and it appears this (or something very similar) is also the convention in the "infinite Ehrenfeucht-Fraïssé/Back-and-forth game" you mentioned in a comment.