I would like to know what is a fair way to split a under water house in a divorce when one spouse choose to stay in the house, we are about 10k to 15k under should the spouse who stays in the house receive any financial compensation from the other spouse who moves out?
That's a question best discussed with your attorney or divorce judge. Another factor to consider is that it will likely be difficult to impossible to refinance that home if it is upside down (depending on what type the existing mortgage is).
Florida is a equitable distribution state - unless the court decides to handle matters differently because of finances or other reasons. This means that if a home is worth $200,000 and paid off, each spouse is entitled to $100,000. If one wants to stay in the house, then that spouse would need to pay the other one $100,000 to release their interest in the real estate.
Assuming you are doing a 50% split and the house is underwater, the court may decide to make the spouse who is leaving the real estate pay off his or her 50% share of the underwater portion or perhaps that person will simply walk away. It really depends on the attorneys and judge and the individual financial situation for each party.