Query for custom posts with a specific tag slug

Since tag is a taxonomy of a custom_post post type, the query could look like this:

<?php
$args = array(
    'post_type'  => 'custom_post',
    'tax_query'  => array(
        array(
            'taxonomy'  => 'post_tag',
            'field'     => 'slug',
            'terms'     =>  array(
                'tag1',
                'tag2',
            ),
        ),
    ),
);

$posts = new WP_Query( $args );

See WP_Query Taxonomy Parameters.