I'm trying to protect a resource in tomcat so that only "valid users" (those with a valid login and password in the realm) can access it. They do not necessarily belong to a group in the realm.  I have tried with many combinations of the <security-constraint> directive without success.  Any ideas? 
Besides the auth-constraint you are adding to the security-constraint:
   <auth-constraint>
       <role-name>*</role-name>
   </auth-constraint>
you need specify the security role in the web-app:
    <security-role>
        <role-name>*</role-name>
    </security-role>
                        There are several realm implementation in tomcat - memory, database, JAAS and more. The easiest one to configure (though not the most secure) the memory one, which contains a single XML file, usually under conf/tomcat-users.xml:
<tomcat-users>
  <user name="tomcat" password="tomcat" roles="tomcat" />
  <user name="role1"  password="tomcat" roles="role1"  />
  <user name="both"   password="tomcat" roles="tomcat,role1" />
</tomcat-users>
The realm configuration is under the context, host or engine configurations, like this:
<Realm className="org.apache.catalina.realm.MemoryRealm"
       pathname="conf/tomcat-users.xml" />
Then, in the web.xml you put the following definition:
    <security-constraint>
            <web-resource-collection>
                    <web-resource-name>MRC Customer Care</web-resource-name>
                    <url-pattern>/protected/*</url-pattern>
            </web-resource-collection>
            <auth-constraint>
                    <role-name>role1</role-name>
            </auth-constraint>
    </security-constraint>
    <!-- Define the Login Configuration for this Application -->
    <login-config>
            <auth-method>DIGEST</auth-method>
            <realm-name>YOUR REALM NAME</realm-name>
    </login-config>
    <security-role>
            <description>
              The role that is required to access the application. 
              Should be on from the realm (the tomcat-users.xml file).
            </description>
            <role-name>role1</role-name>                  
    </security-role>
The web.xml part is taken (with slight change) from one of our web apps.
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