Though I'd be inclined to disallow sloppy markup like this (as others have suggested), one option I've used in the past is to backtrack up the stack looking for a matching tag and autoclosing any open tags I pass along the way.

In this case that would proceed something like this. You get to the </B> and look at the tag at the top of the stack. It's not a <B>, it's an <I>, so you generate a </I> yourself and pop that off the stack, then try again. This time it is a <B> so you can just pop it off the top and you move on.

The next closing tag is </I>. Since there's no matching open tag on the stack, you simply remove it.

        $perlmonks{seattlejohn} = 'John Clyman';