Sounds like she is acting out in public because of the way her daddy handles it. Either he is too permissive and she doesn't have good boundaries, or she gets her way when she does it and she thinks it's how to get her way.
If I were your husband, I would stop whatever it is that they are doing when she starts throwing a fit, take her to the car, and bring her home and put her in her room. The only way she will stop doing this is to NEVER get her way, and pay a big consequence for it.
My older son did this kind of thing. I never let it slide. Even with a cart full of groceries. Even when that meant no movie for my family. We brought home to bear with my very stubborn son that he was NOT going to get his way when he acted like this. It worked.
Your husband needs to sit down with her and tell her the rules. You need to look at her with tough mommy eyes and face and tell her that she had BETTER behave for daddy. Remind her that the moment she stops listening and starts throwing fits, she will lose her privilege. And no matter what, he drops what they are doing and marches her straight to the car.
Better do it now and nip this stuff in the bud. The next thing you know, she will be doing it with YOU when you have the baby with you and then it's very hard to manage both her and the baby.