Actually let me take this back, and file a different bug.

This bug will break too much stuff and is probably not a good way to fix this problem. :)

Filed https://bugs.chromium.org/p/chromium/issues/detail?id=660619

Sample in #1 isn't editable. It seems editability doesn't affect this issue.

 Issue 660619  has been merged into this issue.

I admit that I'm only 80% certain, but this looks like a by design to me. If you think I'm wrong, it'd be helpful to point me a link to spec that supports your expectations.

dir="auto" resolves to the first character, so LTR in this case. And since dir attribute inherits, the base direction of the 2nd div is still LTR.

From my best guess, what you want for your use case is probably not dir="auto", but "unicode-bidi: plaintext":
https://drafts.csswg.org/css-writing-modes-3/#valdef-unicode-bidi-plaintext

Remove dir="auto", and add this CSS (unicode-bidi does not inherit):
  div {
    unicode-bidi: -webkit-plaintext; /* WebKit does not support unprefixed yet */
    unicode-bidi: plaintext;
  }

But I recommend you to double-check the expected behavior with more bidi experts. dir=auto is designed this way intentionally, because that's what authors want in most cases. "unicode-bidi: plaintext" exists to support other, rarer use cases, and I can't determine which behavior is more appropriate for your use case from the description.

In most cases, the base direction is expected to be given explicitly. In case it is not easy, dir=auto gives so-so good heuristics. "unicode-bidi: plaintext" flips to a different heuristics logic for a specific case.

But explicit base direction is always most desired, and using "unicode-bidi: plaintext" will ignore it, so please use it in caution.
https://www.w3.org/International/articles/inline-bidi-markup/uba-basics#base_dir_scope