Akah123,I have no first hand experience of being a second wife or stepmother, so can't offer much insight, just sympathy.
First thing I'd say is that our two children (7 and 2) definitely have a different relationship with mummy and daddy. The oldest says mummy is kinder to him. Even the youngest, at 2 years old, seems to think he can get away with more when he's with me (despite my very best efforts!). When mummy can't make him eat his supper, daddy often can. I assure you I do lots of disciplining as well. Looking around me, this difference in perception ( good cop/ bad cop) seems to be quite common. What I'm getting at is that this girl may always relate to each of you a little differently. It's not just because she is with one 'real' parent and one who is not.
Does your dh leave all the disciplining to you? If so, as I'm sure you know, that's bound to make this girl feel she has to behave when she is with you alone.
Also, she may well be enjoying playing one of you off against the other. Our 7 year old does this. Nothing pleases him more than daddy telling off mummy for not telling off him (or visa versa). It's soooo entertaining and it really lets ds off the hook! Dh and I try really hard not to do this in front of ds. But what's more, if one of us tells off the other about a discipline matter in front of our son, it really chips away at ds's respect for us, making it even more likely he will play up again.
So if you find yourself out with dh and his daughter is misbehaving, I'd say try to avoid any disagreements on discipline in front of his daughter. That's really hard for you, though, because it means tolerating behaviour that stresses you out.
I'm sure you've talked to your husband at great length, so you may have covered all this, but here are some suggested tactics:
Tell your husband that he is doing his daughter absolutely no favours by ignoring her bad behaviour. If she has little respect for other adults, she could have problems fitting in at school and leaning properly. She is just reaching school age, after all. If she's rude and uncontrollable in public, her friends parents will not be very impressed, and she may not get invited to parties and sleepovers etc.
If this falls on deaf ears, you could simply state that much as you love them both, his daughter's bad behavior when you are out together causes you too much stress, so if you reach crisis point, you will have to part company with them for a few hours. If you actually do this, it might shock them both into taking your needs into consideration too. And if your husband relies on you to discipline, a little time alone with his misbehaving daughter may alter his views on the necessity of control.
Good luck!