Does anyone know if this is something we can feasibly address? 

I'm not sure. I *think* it's feasible but also tricky to get right. If we did it with android task management, we'd probably have to keep separate history stacks per instance of the activity. Alternatively, we need to track how many history events are created per instance and when that reaches 0, hit the android back instead of continuing to navigate the backstack.

Note that many native apps also break this, sometimes intentionally sometimes not. For example, IIRC a deep link into a twitter post into the native app, then hitting back takes you to the feed (because Twitter wants to keep you engaged in their app).

Ya, FullScreenActivity.handleBackPressed is pretty naive: if there's a history item, it goes back, regardless of what point in history brought the activity forward.

That said, the flow mentioned in OP doesn't seem right either (or at least after another back). If you're switching to "GSA" that's switching tab stack. So I think you actually have:

Task A
-----
1. Login screen. (Twitter PWA)
2. Feed. (Twitter PWA)
3. Tweet. (Twitter PWA)
4. Profile. (Twitter PWA)
5. Another profile (Twitter PWA)

Task B
------
1. Google Search (GSA)
2. Obama twitter (Twitter PWA) <-- you're here.

(7) Now hit the back button.

I agree that you should get to GSA. But hitting back again, should take you to the homescreen and not back to the PWA (at least I think that's how Android works :) The one tricky bit for us, is that we'd want to navigate the history state back but not actually reload the page until the user comes back to the page.

Anyway, we can put this on the polish and parity list. Thanks for raising it.

Peter is interested in this

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17

commit 5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17
Author: Peter Kotwicz <pkotwicz@chromium.org>
Date: Sat Jul 22 04:27:44 2017

[Android WebAPK] Clear back history when navigating WebAPK from deep link

The CL does this by plumbing LoadUrlParams::should_clear_history_list to Java

BUG= 735074 

Change-Id: I5a7c42abc70b7a6f6175c53156faae9a988faee0
Reviewed-on: https://chromium-review.googlesource.com/573115
Commit-Queue: Peter Kotwicz <pkotwicz@chromium.org>
Reviewed-by: Xi Han <hanxi@chromium.org>
Reviewed-by: Dominick Ng <dominickn@chromium.org>
Reviewed-by: Ted Choc <tedchoc@chromium.org>
Cr-Commit-Position: refs/heads/master@{#488853}
[modify] https://crrev.com/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17/chrome/android/java/src/org/chromium/chrome/browser/tab/Tab.java
[modify] https://crrev.com/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17/chrome/android/java/src/org/chromium/chrome/browser/webapps/WebappActivity.java
[modify] https://crrev.com/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17/chrome/browser/android/tab_android.cc
[modify] https://crrev.com/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17/chrome/browser/android/tab_android.h
[modify] https://crrev.com/5f8fd9e4d2185d6c48f592f4b9f05de1ca04ac17/content/public/android/java/src/org/chromium/content_public/browser/LoadUrlParams.java