I need advice please. 

My boyfriend's 8 y/o daughter has behavior issues that are having a huge negative impact on my 5 y/o daughter and our relationship. 

His 8 y/o daughter and 10 y/o son come over every other weekend, at the same time my 5 y/o is here. At first, things were perfect. We got along with our exes and with each other's ex as well. All the kids got along.

Lately his daughter has been acting out constantly. She refuses to follow any rules. She rebels against any consequences he tries to give. She's extremely disrespectful and even tells my daughter to ignore anything I say. She screams at her father, whines and throws public tantrums where she will refuse to leave a store unless she gets what she wants. She lies consistently, steals/hides my personal items (makeup, nail polish, cell phone), demands constant attention/entertainment from us, whines and tantrums hard when she doesn't get her way and then she demands that her father give her what she wants (or else she won't end the tantrum)..

He admits that he's had "guilty dad syndrome" and for a long time he could not say no to her, so he gave in to her every demand. (My ex used to do this also, but after talking to him about it, he realized the negative effects and has stopped doing it). He's been trying to be consistent, as he has no problem parenting my daughter,  yet when it comes to his own, he cannot follow through with any rules/consequences. 

The discrepancy is a huge issue. I'm left on my own to be the bad guy. Either I say something about her behavior and then she throws a hateful fit, or I ignore the discrepancy and my daughter immediately calls me out in it. "How come she can do it, but I can't?!" His daughter will purposely test this, and as soon as I tell my daughter not to do something, she will sprint across the yard to do the exact same thing, just to rub it in my daughter's face and see if I will say something. 

She lies to her mother and says that we are "mean to her" and that my daughter can do whatever she wants, when in reality it is quite the opposite. She sends horrific "crying face selfies" to her mom anytime someone says no to her. She is overly dramatic and manipulative in trying to get her way in every situation. She is playing her parents against each other. 

My daughter looks up to her and thinks the world of her. After she leaves, my daughter mimics her behavior, and for a week I have to be extra hard on my daughter, just to get rid of the behaviors she's learned over the weekend. It makes me feel terrible.

It's ruining my relationship with my own daughter and causing so much stress and sadness. I dont know what to do. We've talked about this and tried to correct it, but nothing is working. 

Any advice is appreciated.