When John King stopped making payments on his home in Coral Gables, Florida, two years ago, he assumed the foreclosure ended his mortgage contract, he said. Last month, a Miami-Dade County court gave collectors permission to pursue him for $44,000 stemming from the default. King is among a rising number of borrowers who are learning that they can be on the hook for years after losing their homes. Amid a crisis that stripped $6.4 trillion, or 28 percent, from the value of U.S. residential real estate since the 2006 peak, lenders are exercising their rights to pursue unpaid mortgage balances. To get their money, they can seize wages, tap bank accounts and put liens on other assets held by debtors.