Working in Bahrain without Employment Visa/Work Permit

I am not clear what you mean by "deal with this situation"?

There is no dealing with this situation as there are only two options:

1) She gets on the visa of the employer
2) Employer approaches LMRA and asks for work permit/permission for her to work (while on dependent visa).  If approved, her sponsor will remain her husband but employer will be the University

These are the only two ways.  I would NEVER advise her to sign an undertaking and work on dependent visa.  If she is caught, because of the fact that she signed the undertaking, it is clear that she knew that she was doing something illegal and in that case, the maximum penalty will be applied i.e. deportation.

Thanks for the swift response, I asked ‘how to deal with the situation’ because she don’t want to loose her job and at the same time she don’t want to deposit her passport with the employer. Another employee working there has conveyed her that the HR won’t release passport even if the University Ops grant an interim leave to go home for something important for us.
I will advise my friend to convince HR somehow to a get work permit .

Well she can give the passport for visa stamping but then if they don't give it back, she has to go and complain to the police to get it back (as it's not legal for employer to hold the passport).  In any case, the job will be at stake if they don't agree.