Teksti Reader uses Teksti's web fiction archive to display their archives in an embedded form. It offers a unified navigation interface, preloads subsequent pages when reading them in order and allows updating bookmarks on Teksti directly.
This feature is still under development. Hopefully the buttons on the Reader bar are self-explanatory enough. The Reader won't know it if you navigate away within the embedded web fiction. More options for controlling how the content is displayed are available from a menu available in the top left corner.
To try it out, you'll need to select a web fiction. Go back to the main site and select Reader from a web fiction's info page.
Click "Archive" to see a list of web fiction's pages. Clicking a row there takes you that page.
Select a web fiction by clicking "My Web fictions" button. You can move your bookmark in the archive dialog by clicking an already selected row. The "Next" button automatically updates the bookmark if the "autoupdate" option is selected.
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A bit longer explanation on what's going on: Browsers generally disallow embedding insecure content in a secure context. That's why Reader uses HTTP as its protocol instead of HTTPS as the rest of the site does, to avoid browsers' mixed content protections. But that leads to another browser security policy, third party cookies. Accessing your bookmark data with HTTPS is considered such use. Teksti with HTTP is a different thing from Teksti with HTTPS, as far as that's considered.