When can the "build a wonder stage for free" tokens be used?

In the Seven Wonders - Babel Expansion, when can the reward tokens that allow a player "to construct a Stage of their Wonder for free" be used? Can they be used immediately or do they just remove the cost (except coin costs, as noted) but still need to be played on a normal turn by discarding a card?

The question came up during a recent game when a player with Rhodes (B-side), who had participated in the successful project that awarded these tokens, wanted to immediately complete a stage, thereby giving them military dominance over a neighbor. Without finding anything explicit in the rules, we denied the request, supposing that to build a stage the player must have a card in hand to discard and place under their board for that stage. I'm hoping someone else has a ruling that can help.

You have to also use a card, as in the base game. In order to understand it better, look at the previous token, "construct a building for free".

The player takes from the reserve a corresponding token. They can return this token to the reserve to construct a Building for free.

In order to construct a building, you have to play it from your hand, right? So, next time you construct a building from your hand, you may throw in the token and declare "I am constructing it for free". You can't obviously construct immediately an additional building for free, as your hand would have less cards than the other players and thus break the game's mechanism. It works exactly like Olympia's A 2nd wonder bonus.

Same story for the wonder token:

The player takes from the reserve a corresponding token. They can return this token to the reserve, to construct a Stage of their Wonder for free. Note: if the cost of the Wonder stage contains 1 or more coins, these must still be paid.

In order to construct a wonder, you have to put a card below the corresponding wonder stage, to indicate that it has been built (and to remove a card from the game, as well). Thus, you keep the token and the next time you construct a wonder stage you may discard the token to ignore the resource cost. If you could immediately construct a wonder stage, how would you indicate that the wonder has been constructed? A basic principle of the game is that no memory is required; thus, you have to use a card. But you can't just take an extra card from your hand, as you'd leave your hand with less cards and break the game's mechanism.

TL;DR: you keep the wonder token and the next time you use a card to build a wonder stage, you may discard the token to ignore the resource cost of your wonder.

While it’s missing in the English Babel rules, it’s explicitly stated in the German Babel rules (German PDF, page 7, bold emphasis mine):