Here are the two solutions I finally found. Thanks to @Barmar !
- Only with joins:
SELECT
g1.community_id,
g1.publication_name,
g1.publication_name,
g1.publication_id,
COUNT(*) AS rank
FROM
(
SELECT
publications.publication_id as publication_id,
publications.name as publication_name,
publications.community_id as community_id,
communities.name as community_name,
IF(
publications.scheduled_at is not null,
publications.scheduled_at, publications.created_at
) as effective_publishing_date
FROM
feed_items
JOIN publications ON feed_items.publication_id = publications.publication_id
JOIN communities ON publications.community_id = communities.community_id
WHERE
feed_items.user_id = 489387
) AS g1
JOIN (
SELECT
publications.publication_id as publication_id,
publications.community_id as community_id,
IF(
publications.scheduled_at is not null,
publications.scheduled_at, publications.created_at
) as effective_publishing_date
FROM
feed_items
JOIN publications ON feed_items.publication_id = publications.publication_id
WHERE
feed_items.user_id = 489387
) AS g2 ON (
g2.effective_publishing_date, g2.publication_id
) <= (
g1.effective_publishing_date, g1.publication_id
)
AND g1.community_id = g2.community_id
GROUP BY
g1.publication_id,
g1.community_id,
g1.effective_publishing_date
ORDER BY
g1.community_id,
rank ASC;

- With SQL variables
SELECT data_table.publication_id, data_table.publication_name, data_table.community_id, data_table.effective_publishing_date,
@publication := IF(@community <> data_table.community_id, concat(left(@community := data_table.community_id, 0), 0), @publication+1) AS rank
FROM
(SELECT @publication:= -1) p,
(SELECT @community:= -1) c,
(SELECT
publication.name as publication_name,
publication.community_id as community_id,
feed_item.publication_id as publication_id,
IF(publication.scheduled_at is not null, publication.scheduled_at, publication.created_at) as effective_publishing_date
FROM feed_items feed_item
JOIN publications publication ON feed_item.publication_id = publication.publication_id
WHERE feed_item.user_id = 489387
ORDER BY publication.community_id, effective_publishing_date ASC
) data_table;