A dream sparked something inside my mind, and I told my little-sister-and-fellow-author all about it. I told her that I now wanted to write a fantasy story, and I kept coming back to tell her the character ideas I was getting. I tried out different concepts on her, absorbed her advice, and let the story brew. Finally I started to write.

Later, she started her own fantasy story. (Inspired by this forum!) She told me all about her characters, tried out different ideas on me, and started to write.

We continued like this for quite a while, thinking, plotting, developing our characters and building our separate worlds. Then, one day, she told me how she had changed several elements of her world---and I realized that our worlds were now very similar.

"Hey," I said to her, "do you want to merge our worlds?"

She was interested in the idea, so I pulled out an old notebook and scribbled a rough map of my world. Then she did the same. We rearranged several spots, until we had merged our countries (we had only two countries each) into a nice continent. We started merging our histories (which before this had been very vague) until we had a very tidy timeline. We held world building sessions, and the ideas came fast, unlike when we were trying to build solo. And we liked our history so much that we want to write a prequel.
In short, it was a success, and very, very fun. We each gained two new countries, two new nationalities, a much improved history, and inspiration to write more.

Do you know anybody with an incomplete world that is similar to your incomplete world?

I haven't experienced this myself ... but this sounds very much like what C.S. Lewis and his brother did as children. Lewis's "Animal-Land" (which was superficially similar to Narnia---talking animals---but at a deeper level not nearly so interesting) and his brother's "India" merged to form "Boxen", which is also the title of the book in which several of the fragments were published. Not that those two imaginary countries were really all that similar (except that they sprung from the imaginations of two brothers), but the merging process ...

That's cool. I didn't know about Boxen.

Oh, and by "similar", I didn't mean that they look a lot alike. I meant that they have the same rules. For example, in the beginning, my story had only human people (which was very important to my world's moral system), while my sister's had elf and mermaid people. But eventually she dropped the elves and mermaids, so we were able to merge. And neither of us had magic in our worlds, which made them compatible.
Furthermore, the countries your stories take place in don't even have to know about one another. But merging helps both authors in building a world, which can be a daunting task.

This isn't exactly the same sort of thing, but one day Arien and I were talking about my stories and realized that a bunch of them took place in the same world. Ever since then, Arien's helped me develop things and fit them together. It's all my world, but he definitely has a share in it, and has helped me make it so much better than it was before.

Arien and I are also working on developing a "multiverse" together. For that one, it doesn't matter that our worlds have different rules; we just have to figure out how things change when people/things move from one universe to another.

A multiverse? That sounds interesting. Could you further explain?

Basically, there's more than one universe (uni = one, multi = more than one). Each universe can have multiple worlds, and each universe has its own set of rules. The universes are linked, though, and there's at least one place that has a portal to one world in each of the other universes.

Usually with multiverses, the universes are more parallel to each other, with a lot of things being similar, or potentially sharing some history and people. With our multiverse, each universe is very much its own place, with its own rules, characteristics, and history.

Honestly, this won't affect our stories much. There isn't a lot of connection between the universes, although one of my characters has a sword that came from Arien's universe. Mostly it's just a fun little thing for us to be aware of, and maybe someday we'll write a story where some of our characters meet each other.

That is fun.
Co-world building is so much fun. I love swapping ideas with my sister; a couple times (many times) we have talked late into the night about stories, and authorship. We read and critique each other's stories, and poke each other to keep writing. Merging our worlds has just added to the funness.