Over the years, William R. Ponsoldt had earned tens of millions of dollars building a string of successful companies. He had renovated apartment buildings in the New York City area. Bred Arabian horses. Run a yacht club in the Bahamas, a rock quarry in Michigan, an auto-parts company in Canada, even a multibillion-dollar hedge fund.

Now, as he neared retirement, Mr. Ponsoldt, of Jensen Beach, Fla., had a special request for Mossack Fonseca, a Panama-based law firm well placed in the world of offshore finance: How could he confidentially shift his money into overseas bank accounts and use them to buy real estate and move funds to his children?…[+]