Shariah-compliant banking, sometimes called Islamic banking, is growing in popularity in the Western and Islamic worlds. But critics say American interest in the system at a time of economic crisis is opening the door to increased Islamic influence in the American banking system. Worse yet, some fear the banks may be helping to finance international terrorism. In Shariah-compliant banking, lenders may not charge interest and investors cannot make money from forbidden industries like gambling, alcohol, pork and pornography. Selling debt, devising derivatives and short selling are also prohibited, and investments must be closely tied to actual assets. In the U.S.,...