When reading anonymously in a browser on Twitter, there comes a point when Twitter forces you to login in order to continue.
Is there a way to continue without needing to login?
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In order to browse Twitter in your web browser without restrictions, you must login when prompted, or start another session.
In order to access Twitter anonymously, you will need to get the data through a method other than their website. There are a number of approaches people use, some are pay-to-use, some not.
For example you could take a look at nitter project on Github which does the anonymizing for you.