 Product forcing

The conditions are pairs with coordinatewise extension: $(p,q)\le (p^{\prime },q^{\prime })$ exactly when $p\le p^{\prime }$ and $q\le q^{\prime }$. The projection of a product-generic filter is a pair of factor filters whose Cartesian product is the original filter. Each is ground-model generic, and mutual genericity for product forcing makes the second generic over the extension by the first.