Posts pagination with custom SQL

Hello,

I have a really weird problem. I have a loop in archive.php that displays posts in a custom way. When I access to it http://domain.ext/event/ (this is a custom post type), it works fine. Here the generated query:

SELECT SQL_CALC_FOUND_ROWS $wpdb->postmeta.meta_value AS event_date, $wpdb->posts.*
FROM $wpdb->postmeta
RIGHT JOIN $wpdb->posts ON ($wpdb->posts.ID = $wpdb->postmeta.post_id)
$wpdb->postmeta.meta_key = 'event_date'
WHERE $wpdb->postmeta.meta_value >= NOW()
ORDER BY $wpdb->postmeta.meta_value ASC
LIMIT $start, $posts_per_page

But when I want to see a specific category, pagination doesn’t work anymore (http://domain.ext/category/a-category/):

SELECT SQL_CALC_FOUND_ROWS $wpdb->postmeta.meta_value AS event_date, $wpdb->posts.*
FROM $wpdb->postmeta
RIGHT JOIN $wpdb->posts ON ($wpdb->posts.ID = $wpdb->postmeta.post_id)
$wpdb->postmeta.meta_key = 'event_date'
RIGHT JOIN $wpdb->term_relationships ON ($wpdb->posts.ID = $wpdb->term_relationships.object_id)
RIGHT JOIN $wpdb->term_taxonomy ON ($wpdb->term_relationships.term_taxonomy_id = $wpdb->term_taxonomy.term_taxonomy_id)
WHERE $wpdb->term_taxonomy.taxonomy = 'category'
AND $wpdb->term_taxonomy.term_id IN ($category_id) AND
AND $wpdb->postmeta.meta_value >= NOW()
ORDER BY $wpdb->postmeta.meta_value ASC
LIMIT $start, $posts_per_page

Here is my (simplified) template:

<?php
$category_id = get_query_var( 'cat' ); // Used in the query.

$posts_per_page = get_option( 'posts_per_page' );

$start = ( get_query_var( 'paged' ) - 1 ) * $posts_per_page;
$start = $start >= 0 ? $start : 0; // We need a positive number.

$sql = "SELECT SQL_CALC_FOUND_ROWS $wpdb->postmeta.meta_value ...";

if ( is_category() ) {
  $sql .= "RIGHT JOIN ...";
} else {
  $sql .= 'WHERE ';
}

$sql .= "$wpdb->postmeta.meta_key = 'event_date' ... LIMIT $start, $posts_per_page";

$posts = $wpdb->get_results( $sql, OBJECT );
?>

<?php if ( $posts ) : global $post; ?>

  <?php foreach ( $posts as $post ) : setup_postdata( $post ); ?>

    <?php get_template_part( 'loop' ); ?>

  <?php endforeach; ?>

  <?php get_template_part( 'posts-pagination' ); ?>

<?php endif; ?>

I also tried to manually set max_num_posts and found_posts in $wp_query, pagination appears, but I get a 404 error.