Multiple relationship for multiple tax_query in WP_Query

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This can be done by using the term_taxonomy_id rather than the slug, which will effectively ignore whichever taxonomy is specified and just look at the unique term_taxonomy_id field. This will allow you to effectively be able to do a mixed relationship. You’d want to use an overall relation of AND and put all the terms that should be related OR in one item using the IN operator. You will need to map the terms desired to their term_taxonomy_ids first.

$taxes = array( 'taxonomy1', 'taxonomy2', 'taxonomy3', 'taxonomy4' );

foreach ( $taxes as $tax ) {
    $terms = get_terms( $tax );

    foreach ( $terms as $term )
        $tax_map[$tax][$term->slug] = $term->term_taxonomy_id;
}


$args['tax_query'] => array(
    'relation' => 'AND',
    array(
        'taxonomy' => 'taxonomy1',
        'field' => 'term_taxonomy_id',
        'terms' => array( $tax_map['taxonomy1'][$slug] )
        'operator' => 'IN',
    ),
    array(
        'taxonomy' => 'taxonomy3',
        'field' => 'term_taxonomy_id',
        'terms' => array( $tax_map['taxonomy3'][$slug] ),
        'operator' => 'IN',
    ),
    array(
        'taxonomy' => 'taxonomy4', // gets ignored
        'field' => 'term_taxonomy_id',
        'terms' => array( $tax_map['taxonomy4'][$slug], $tax_map['taxonomy2'][$slug] ),
        'operator' => 'IN',
    ),
);

Note that pre-3.5, you’d also need to specify 'include_children' => false. See this Trac ticket for more: https://core.trac.wordpress.org/ticket/21228

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