Lots of compliments like 'I really admire you when you stand your ground and are consistent about boundaries' (in your own words, of course) - gives the message without it looking too much like interfering. Shouting at each other isn't on, but how can you discipline the child when mum is doing it?
I found that talking in a quiet but stern voice, like a broken record - 'we are not discussing this while there is shouting, let's take 10 minutes out' until they got the message, was helpful. If he shouts at mum and she is being appropriate, he could have privileges removed, be grounded etc, for a reasonable time that is achievable. Telling them 'no TV' when you're looking forward to relaxing for the evening in front of the TV is bound to fail.
I imagine she is frustrated with this, too. My daughter was tearing her hair out with her hyper 3.1/2 year old a couple of weeks ago - on Easter Sunday, they were sugared up to the hilt, had missed a lot of sleep whilst staying over with their dad, were really pushing the boundaries and running off when she wanted to speak quietly to them. Just talking it through and being prepared for when it would happen again did the trick - they knew she was ready to deal with them, and behaved like angels!
I hope your daughter can collaborate with you and any other adult that is present to see this behaviour, so they can all be consistent. But it's common behaviour for children - they just have to learn they won't get away with it.
My adult children now acknowledge that they knew when I meant it, and when they thought they'd get away with a bit more. It was the 'look' of determination and knowing I had the power to withdraw privileges.