Wordpress - Related Posts by Multiple Tags?

I had the same idea and wrote a small little plugin to help me do this.

function get_pew_related_data($args, $post_id, $related_id) {
    global $post, $wpdb;
    $post_id = intval( $post_id );
    if( !$post_id && $post->ID ) {
        $post_id = $post->ID;
    }

    if( !$post_id ) {
        return false;
    }

    $defaults = array(
        'taxonomy' => 'topics',
        'post_type' => array('post'),
        'max' => 5
    );
    $options = wp_parse_args( $args, $defaults );

    $transient_name = 'pew-related-' . $options['taxonomy'] . '-' . $post_id;

    if( isset($_GET['flush-related-links']) && is_user_logged_in() ) {
        echo '<p>Related links flushed! (' . $transient_name . ')</p>';
        delete_transient( $transient_name );
    }

    $output = get_transient( $transient_name );
    if( $output !== false && !is_preview() ) {
        //echo $transient_name . ' read!';
        return $output;
    } 

    $args = array(
        'fields' => 'ids',
        'orderby' => 'count',
        'order' => 'ASC'
    );
    $orig_terms_set = wp_get_object_terms( $post_id, $options['taxonomy'], $args );

    //Make sure each returned term id to be an integer.
    $orig_terms_set = array_map('intval', $orig_terms_set);

    //Store a copy that we'll be reducing by one item for each iteration. 
    $terms_to_iterate = $orig_terms_set;

    $post_args = array(
        'fields' => 'ids',
        'post_type' => $options['post_type'],
        'post__not_in' => array($post_id),
        'posts_per_page' => 50
    );
    $output = array();
    while( count( $terms_to_iterate ) > 1 ) {

        $post_args['tax_query'] = array(
            array(
                'taxonomy' => $options['taxonomy'],
                'field' => 'id',
                'terms' => $terms_to_iterate,
                'operator' => 'AND'
            )
        );

        $posts = get_posts( $post_args );

        /*
        echo '<br>';
        echo '<br>';
        echo $wpdb->last_query;
        echo '<br>';
        echo 'Terms: ' . implode(', ', $terms_to_iterate);
        echo '<br>';
        echo 'Posts: ';
        echo '<br>';
        print_r( $posts );
        echo '<br>';
        echo '<br>';
        echo '<br>';
        */

        foreach( $posts as $id ) {
            $id = intval( $id );
            if( !in_array( $id, $output) ) {
                $output[] = $id;
            }
        }
        array_pop( $terms_to_iterate );
    }

    $post_args['posts_per_page'] = 10;
    $post_args['tax_query'] = array(
        array(
            'taxonomy' => $options['taxonomy'],
            'field' => 'id',
            'terms' => $orig_terms_set
        )
    );

    $posts = get_posts( $post_args );

    foreach( $posts as $count => $id ) {
        $id = intval( $id );
        if( !in_array( $id, $output) ) {
            $output[] = $id;
        }
        if( count($output) > $options['max'] ) {
            //We have enough related post IDs now, stop the loop.
            break;
        }
    }

    if( !is_preview() ) {
        //echo $transient_name . ' set!';
        set_transient( $transient_name, $output, 24 * HOUR_IN_SECONDS );
    }

    return $output;
}

function pew_related( $args = array(), $post_id = '', $related_id = '' ) {
    $post_ids = get_pew_related_data( $args, $post_id, $related_id );

    if( !$post_ids ) {
        return false;
    }

    $defaults = array(
        'post__in' => $post_ids,
        'orderby' => 'post__in',
        'post_type' => array('post'),
        'posts_per_page' => min( array(count($post_ids), 10)),
        'related_title' => 'Related Posts'
    );
    $options = wp_parse_args( $args, $defaults );

    $related_posts = new WP_Query( $options );
    if( $related_posts->have_posts() ):
    ?>
    <h5><?=$options['related_title']?></h5>
    <div id="related-material" class="promo">
        <?php while ( $related_posts->have_posts() ):
            $related_posts->the_post();
        ?>
            <a class="post" href="<?=the_permalink();?>">
                <div class="meta">
                    <?php
                    $post_project = wp_get_object_terms($related_posts->post->ID, 'projects');
                    $project = 'Pew Research Center';
                    $project_slug = '';
                    if( isset($post_project[0]) ) {
                        $project = $post_project[0]->name;
                        $project_slug =  $post_project[0]->slug;
                    } elseif( $related_posts->post->post_type == 'fact-tank' ) {
                        $project = 'Fact Tank';
                        $project_slug = 'fact-tank';
                    }
                    ?>
                    <span class="project <?=$project_slug;?> right-seperator"><?=$project;?></span>
                    <span class="date"><?php the_time('M j, Y'); ?></span>
                </div>
                <h2><?=the_title();?></h2>
            </a>
        <?php endwhile;
            wp_reset_postdata();

         ?> 
    </ol>
    </div>
    <?php
    endif;

}

It looks for posts that have common terms and the terms are sorted by frequency so the least used terms come first then the more popular terms. The first function fetches the data and stores it in a transient so the results aren't run over and over and over again unnecessarily. The second function just renders the output. This is what powers our related posts on one of our sites at work http://www.pewresearch.org/fact-tank/2013/08/02/both-parties-underwater-heading-into-2014-elections/

The algorithm works like this:

1. Get all the terms from the post ordered by count in ascending order (least popular to more popular)
2. Loop over this set of terms and look for posts that contain Term1 AND Term2 AND Term3
3. With each iteration remove the least popular term from the list broadening our results until we get the desired number of posts or we only have one term left to check.
4. If we still don't have enough posts to meet our needs, then look for posts that contain Term1 OR Term2 OR Term3
5. Save the result to a transient so we don't have to run these queries again for a while.

Hope this helps you out.