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I am not entirely sure what you should do. But, at least at many American math departments, I can make two suggestions for what I wouldn't do:

  1. Don't try to keep it secret for as long as possible. Sane math departments want to help with two-body problems. But it takes real work to help, and the department can't take do any of that work until you define the problem. It could be a mistake to be very forward about a two-body problem, as if to suggest that that's supremely more important than research and teaching. One deft move which is I think is fine is to have a link to your partner toward the end of your home page, without an explicit statement that you have a two-body problem. Also, you have to define your two-body problem privately to each university when the time comes; what you say on your home page is not enough. (As Jordan Ellenberg says below, it's outright illegal for the university to infer your romantic status or even ask you.)
  2. Certainly if you are both in math, don't interview on the same plane trip. It's important to establish that you and your partner are independent professionals. The department has the right to schedule you on the same plane trip and I wouldn't say that you should draw a line in the sand if they do. But it's better for both sides to avoid this.
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I am not entirely sure what you should do. But, at least at many American math departments, I can make two suggestions for what I wouldn't do:

  1. Don't try to keep it secret for as long as possible. Sane math departments want to help with two-body problems. But it takes real work to help, and the department can't take do any of that work until you define the problem. It could be a mistake to be very forward about a two-body problem, as if to suggest that that's supremely more important than research and teaching. One deft move which is I think is fine is to have a link to your partner toward the end of your home page, without an explicit statement that you have a two-body problem. Also, you have to define your two-body problem privately to each university when the time comes; what you say on your home page is not enough.
  2. Certainly if you are both in math, don't interview on the same plane trip. It's important to establish that you and your partner are independent professionals. The department has the right to schedule you on the same plane trip and I wouldn't say that you should draw a line in the sand if they do. But it's better for both sides to avoid this.