Is it possible to teach abroad with your significant other?

Definitely! You might not get jobs at the same school, but if you're in a bigger city you should both be able to find work. If either of your schools provide housing, though, just make sure that you're able to find a place that's big enough for the two of you. Some schools will just give their teachers a simple studio but might be willing to give you cash in lieu of a place or something.

I know plenty of people who have taught abroad with a significant other. From what I've seen in China, a school will hire the two of you together and have you share an apartment (a win-win for them!). If you can't find a school that will take you both, try applying for jobs at schools near each other or within reasonable commuting distance. I would steer clear of placement programs unless they can guarantee they can place you two at the same school together, or at schools near each other.

You haven't mentioned if your significant other is a teacher or not. If not, schools will ask if you have dependents and they will sponsor your significant other as a dependent. It all depends on where you want to go. In middle-east, they will sponsor and even pay for half of the traveling expenses of your significant other.

I don't know if you can do so there, but you definitely can in Columbia, Korea, & China.