Tanstack Start Supabase Auth documentation is broken

Which project does this relate to?

Router

Describe the bug

Which project does this relate to?

Start

Describe the bug

This Supabase Auth + Tanstack start is currently half broken.
There seems to be two major problems:

1. In the fetchUser (__root.tsx) server function, it is returned directly data.user that cointains non-serializable values. Here it's sufficient to no return the entire object
2. When the page loads the getSupabaseServerClient function causes an hydration error

Additionally

1. In __root.tsx getSupabaseServerClient does not need to be awaited

Your Example Website or App

https://tanstack.com/start/latest/docs/framework/react/examples/start-supabase-basic

Steps to Reproduce the Bug or Issue

1. Go to the example site page
2. Open it with code sandbox
3. Open the console and reload the page

Expected behavior

There should not be any errors with tsc when cloning the template and there should not be an hydration error when the page loads

Screenshots or Videos

No response

Platform

• OS: [e.g. macOS, Windows, Linux]
• Browser: [e.g. Chrome, Safari, Firefox]
• Version: [e.g. 91.1]

Additional context

No response

Start with the start-supabase-basic example linked in the issue and inspect __root.tsx, especially fetchUser and getSupabaseServerClient. Clone the template, run tsc, then reload the example while watching the console. Done means the type-check passes and the page loads without hydration or serialization errors.