(Sorry for reposting this, since the last post was closed for further comments and I couldn't reply)
This is not an Android issue - see two screenshots below, one with GBoard, one with SwiftKey - both of them work fine for the first language (English), but for the second language (Hebrew), SwiftKey shows all words as misspelled. For GBoard, I also added a misspelled word at the end of the Hebrew text, just to show that the spell checker does work there.
The solution @Alpert referred me to here (below), turns off spell check entirely, which is not something I'm looking for - I do want it, but I want it to work properly.
For some reason, SwiftKey Android shows red underlines under *all* words in my second language (Hebrew), like they're all misspelled. It does work fine for the first language (=English).
This issue doesn't happen when I switch to another keyboard.