Anti-fraternizations are hard to enforce. The relationships can be hard to detect and it's easy for people to start saying "what about John and Mary? Everyone knew about them, why are you picking on us?" They're also hard to defend: you have a constitutional right to a freedom of association. That's why most companies have a notification requirement rather than an anti-fraternization requirement. To whatever extent this will vary it will vary on a circuit by circuit basis. My understanding is that the 5th and 8th circuits are very hostile to anti-fraternization policies. I am not aware of any state laws in place.
If you are in a relationship that is important to you, then I would split the cost of an attorney who you will have consider the matter for you. If the policy is unenforceable, I would have the attorney write a letter to the employer stating that (s)he represents two current employees in a relationship and that the company's policy will lead to legal action if the company moves to enforce it against the represented employees. If the attorney feels that (s)he must reveal your names, then you will want to be very clear with your partner about how far you are going to go with this and how strongly you believe your attorney.
EDIT: There is one caveat: if your work involves security clearance, government funding, or if the employer is the government, the rules may be different for you.