The card Rebel Cantina gives you cards during the Produce phase for every rebel world in your tableau. But what if another player plays the Produce? Does it still count?
Yes. The only difference between the player who plays "Produce" and everyone else is the free good on a windfall world. Every other effect applies equally to every player - there's no difference between Rebel Cantina and any other world with a "Produce" effect.
You said it perfectly in your question "during the Produce phase".
When a player plays the V-Produce action card, he iniates that phase for every player, and he gets a bonus for that phase. All cards that have an effect in the V slot are activiated regardless of who iniated the Produce phase.