I am sorry to hear about this challenging situation.
The ideal and example of the Prophet (S) is that marriage is public and, unless someone is living in a cave, there are a lot of questions and awkward situations that will come up if a woman is married secretly.
It is wrong to put someone under that kind of stress and pressure, and it is not healthy for the man or the woman.
Apart from the social challenges, marriage is one of the most important things in a person's life, and it is psychologically unhealthy to have to hide that from the world. Part of having a family is the social aspect of family, not just the private relationship between a husband and wife.
However, unfortunately, life isn't ideal. I am assuming the woman agreed to this situation when getting into it. Maybe - like most things in life - they didn't understand what it would be like long term, or they thought (or were told) things might change.
The only real options are: (a) try to convince the man to change his mind, (b) try to see if it is possible to make any life changes (such as moving to a different country) which would make secrecy unnecessary, (c) find coping mechanisms to deal with the situation personally and socially, and be patient, (d) leave the marriage, or (e) pray that Allah provides another option.
Life is a test, and in the end, what is important is how we navigate the tests in front of us, whether we are able to do it with faith and good ethics or not. I hope and pray the situation gets easier for all concerned!