If you need the old (<4.1) behavior of register_shutdown_function you can achieve the same with "Connection: close" and "Content-Length: xxxx" headers if you know the exact size of the sent data (which can be easily caught with output buffering).
An example:
<?php
header("Connection: close");
ob_start();
phpinfo();
$size=ob_get_length();
header("Content-Length: $size");
ob_end_flush();
flush();
sleep(13);
error_log("do something in the background");
?>
The same will work with registered functions.
According to http spec, browsers should close the connection when they got the amount of data specified in Content-Length header. At least it works fine for me in IE6 and Opera7.