If your husband is tuning out the tantrums he has completely the right idea (and is possibly deaf )Its not easy to do (my 10 year old has tantrums too and I have trouble taking my own advice on this one) but it is the only way I know to successfully get the message through that it is a complete waste of there time behaving like that. Only talk to her about the problem when she is calm.

Do you think maybe as you deal with things your husband thinks theres nothing for him to do? Perhaps taking more of a back seat would inspire him to take more responsibility?

Maybe your husband does not know how to respond to her tantrum, or maybe he is afraid that if he gets involved, he might not be on the same page with you(It happened to us). Talk to him about how he feels when your daughter throwing tantrum and ask him what his approach is if he is the one to discipline her. Regarding your daughter's tantrum, it might be something bothers her making her behave that way. Welcome her to talk about the problem in detail and do not judge her feeling, giving her a ear and show your understanding. Ask her about possible solutions she can come up with the conflict. It will take time. Good luck!