continue processing php after sending http response

Yes. You can do this:

ignore_user_abort(true);//not required
set_time_limit(0);

ob_start();
// do initial processing here
echo $response; // send the response
header('Connection: close');
header('Content-Length: '.ob_get_length());
ob_end_flush();
@ob_flush();
flush();
fastcgi_finish_request();//required for PHP-FPM (PHP > 5.3.3)

// now the request is sent to the browser, but the script is still running
// so, you can continue...

die(); //a must especially if set_time_limit=0 is used and the task ends

I've seen a lot of responses on here that suggest using ignore_user_abort(true); but this code is not necessary. All this does is ensure your script continues executing before a response is sent in the event that the user aborts (by closing their browser or pressing escape to stop the request). But that's not what you're asking. You're asking to continue execution AFTER a response is sent. All you need is the following:

// Buffer all upcoming output...
ob_start();

// Send your response.
echo "Here be response";

// Get the size of the output.
$size = ob_get_length();

// Disable compression (in case content length is compressed).
header("Content-Encoding: none");

// Set the content length of the response.
header("Content-Length: {$size}");

// Close the connection.
header("Connection: close");

// Flush all output.
ob_end_flush();
@ob_flush();
flush();

// Close current session (if it exists).
if(session_id()) session_write_close();

// Start your background work here.
...

If you're concerned that your background work will take longer than PHP's default script execution time limit, then stick set_time_limit(0); at the top.

If you're using FastCGI processing or PHP-FPM, you can:

session_write_close(); //close the session
ignore_user_abort(true); //Prevent echo, print, and flush from killing the script
fastcgi_finish_request(); //this returns 200 to the user, and processing continues

// do desired processing ...
$expensiveCalulation = 1+1;
error_log($expensiveCalculation);

Source: https://www.php.net/manual/en/function.fastcgi-finish-request.php

PHP issue #68722: https://bugs.php.net/bug.php?id=68772

I spent a few hours on this issue and I have come with this function which works on Apache and Nginx:

/**
 * respondOK.
 */
protected function respondOK()
{
    // check if fastcgi_finish_request is callable
    if (is_callable('fastcgi_finish_request')) {
        /*
         * This works in Nginx but the next approach not
         */
        session_write_close();
        fastcgi_finish_request();

        return;
    }

    ignore_user_abort(true);

    ob_start();
    $serverProtocole = filter_input(INPUT_SERVER, 'SERVER_PROTOCOL', FILTER_SANITIZE_STRING);
    header($serverProtocole.' 200 OK');
    header('Content-Encoding: none');
    header('Content-Length: '.ob_get_length());
    header('Connection: close');

    ob_end_flush();
    ob_flush();
    flush();
}

You can call this function before your long processing.