There is no support for the very creative sql statement that you have constructed.
There may be other ways, but for expediency, best is to use a loop:
UPDATE interns i, intern_complete_application ica SET i.is_complete = ica.is_complete WHERE i.id = ica.intern_id
$result=mysql_query("SELECT intern_id FROM intern_complete_application WHERE is_complete = 1");
//NOTE: I don't know what you're using for 'is_complete'
while($row=mysql_fetch_array($result)){
mysql_query("UPDATE interns SET is_complete =1 WHERE id=$row[0]");
}