If a woman asks her husband for a divorce so she can marry another man, and he says to her, “You are divorced,” and then she regrets it and claims it was a joke, is this considered a real divorce that necessitates a new marriage contract?

Since you uttered a clear divorce, it has taken effect. If you had not consummated the marriage with your wife or been in a valid seclusion with her, then she is irrevocably divorced from you with that single pronouncement, and you do not have the right to take her back except with a new contract. However, if you had consummated the marriage with her or been in a valid seclusion with her, then you have the right to take her back before the expiration of her waiting period without the need for a new contract, provided that this divorce does not complete the three pronouncements.