Name for a certain “product game”

Let $G,H$ be two (combinatorial impartial) games. Consider the following new game $P$: The positions are the pairs of positions of $G$ and $H$. A move in $P$ is a move in $G$, or a move in $H$, or a move in both. If we end in a terminal position of $G$, we go on and play in $H$. Similarly with $H$ and $G$.

What is the name of this "product game"? Has it been studied somewhere? In particular, what are the $\mathcal{P}$-positions of $P$ in terms of the ones for $G$ and $H$?

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In Winning Ways, this is called the "selective compound" of $G$ and $H$.

In general, the selective compound of a finite number of games is the game where you choose some non-empty subset of the component games at each turn and make one move in each of the chosen games.

In the impartial case, the selective compound is P if all the component games are P (because your opponent can respond in whatever subset of games you move in), and otherwise it's N (because you can move from N to P in all the N components).

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Perfect! Thanks a lot. –  Martin Brandenburg Apr 26 '12 at 7:48