I’ve heard this phrase quite a few times (mostly in anime) but couldn’t quite catch the meaning. I’m not sure if it can be translated directly into English, but someone gave me a good example of when to use it:
When, for example, you sugest something like 飲みに行こうよ! but the person you are talking to constantly refuses. When that person eventually agrees, you can use そうこなくちゃ in delight. It seems to be used when there is some opposition to what you want to happen at first before you get your own way.