In the replay details, I can see a network tab, but it only includes some HTTPS requests. Can I also get information about WebSockets there?

By default, Session Replay will capture basic information about all outgoing fetch and XHR requests in your application. At the moment, WebSocket connections will not be captured. Please see our docs here on Session Replay Network Details.

As an alternative, you can capture more information using the CaptureConole integration, which captures all Console API calls. Another option is to use Sentry.captureException, which manually reports errors.