My kids are completely different people with me than with my wife too. Are you the disciplinarian in your house? I am in mine and I have no problems at all when my wife isn't around. When she is, the kids will throw tantrums, cry over nothing, whinge, whine and complain and generally be a nuisance.
D4 is the worst and throws tantrums as you described. I've taken to ignoring her behaviour and when I can get a word in, I explain to her that I will not talk to her until she settles down. I leave her to have her moment, I'll move her to her bedroom is she is disrupting what anyone else is doing, and I'll go about my business as if nothing were happening. When she finishes, I'll say my piece, she'll be cranky for a while and then become very affectionate later on when she calms down.
Me: 31, W: 29 T: 4 M: 2 Kids: 3 (SS: 7, SD: 4, D: 3) Separated, still living together: Nov 2013 Separate bedrooms: Feb 2014 W working away; kids with me: Nov 2014