What causes "java.lang.SecurityException: Failed to authenticate principal=null, securityDomain=jmx-console" in JBoss EAP 4 or 5?

Issue

When invoking operations dependent on JMX MBeans (for example, dm.getAvailableModules(ModuleType.EAR, targets)), exceptions similar to the following are thrown:

15:56:56,703 ERROR [JMXTarget] Cannot get available modules
java.lang.SecurityException: Failed to authenticate principal=null, securityDomain=jmx-console

Or:

java.lang.SecurityException: Failed to authenticate principal=null, securityDomain=jmx-console
      at org.jboss.jmx.connector.invoker.AuthenticationInterceptor.invoke
      at org.jboss.mx.server.Invocation.invoke
      at org.jboss.mx.server.AbstractMBeanInvoker.invoke
      at org.jboss.mx.server.MBeanServerImpl.invoke
      at org.jboss.invocation.jrmp.server.JRMPProxyFactory.invoke
      ...
      javax.servlet.http.HttpServlet.service
      ...