NEW YORK — Reviewers have raved about Batman: Arkham City, the action-packed superhero game that publisher Warner Bros. released on Tuesday for Xbox 360 and PlayStation 3.

One aspect of the Dark Knight’s latest adventure that has found itself the subject of controversy is the way it handles the Catwoman segments of the game. These are unlocked with a “VIP Pass,” a redemption code printed on a sheet of paper inside new copies of the game. Gamers who rent, borrow or buy a used copy of Arkham City will have to pay an extra $10 for the code. And those who can’t connect their consoles to the I...