Maybe taking advantage of https://www.postgresql.org/docs/12/textsearch-controls.html#TEXTSEARCH-PARSING-QUERIES
SELECT websearch_to_tsquery('english', '"sad cat" or "fat rat"');
websearch_to_tsquery
-----------------------------------
'sad' <-> 'cat' | 'fat' <-> 'rat'
(1 row)
Or maybe we should create our own parser with:
or