Don't want a complete HTML document?
Tidy has a wide range of configuration options that allow you to manipulate how the output of tidy will look. In this case, you can use the 'show-body-only' option to keep tidy from creating a full HTML document.

Setting options
<?php
    $tidy = new tidy_doc();
    $tidy->parseFile("intro2_ex1.html", array("show-body-only" => true));
    $tidy->cleanRepair();

    echo $tidy;
?>
Result
<p>Hello, <b>This HTML</b><i>is all sorts of</i> messed up!</p>