Actually, honestly, this is NOT something that happens to everyone. I have an ex husband and a current husband. My ex- although he has his issues, lol- ALWAYS made sure we were on the same page parenting, especially when my son was very small and we were still married. Even now when our son is 11, we call and confer to make sure we both have basically the same stance on things.
My current husband and I also make sure we have a 'united front'. Even if it requires you and your spouse working it out ahead of time, you REALLY need to make rules and BOTH stick to it. If you don't, I can promise you it will come back to bite you later on.
Kids will play both parents off of each other and will use any excuse to get out of 'rules'. It doesn't make them bad- just kids, learning how far they can go and how rules work. By undermining your 'everyday' rules, your partner is teaching your daughter that A) she doesn't have to listen to mama, and B) that 'rules' are meaningless and can be broken whenever she or someone else like daddy, feel like breaking them.
This doesn't seem AWFUL - but tell your husband this: A 3 year old doesn't have the judgement skills to know WHICH rules are a 'big deal' and which are not. So in other words, if he says it is ok to break the 'juice rule' or the 'TV rule' your daughter will not see any difference in also breaking the 'don't cross a busy street' rule or the 'don't talk to strangers' rule. All she is learning is that the CONCEPT of rules is not something she has to obey.
Put it to him like that and see if he 'gets it'. Also, does he want her to go to preschool or kindergarten in a year or two and to think that she does not have to follow 'teacher's rules' like holding hands, staying together, don't push, stay in line, wash your hands, etc?
Kids that young are learning the CONCEPT of rules- and you must be absolutely consistent to make sure she gets that and can be safe and happy in public. All these good and safe kiddie behaviors start at HOME. Your partner totally needs to get over himself and be a better parent and think about the bigger picture of how this stuff impacts his child.