[bug] localStorage not syncing between multiple windows on Linux

[The-Best-Codes]

Describe the bug

localStorage, sessionStorage, etc. is not synced between two instances of an app, even if the window labels are the same. This is a large problem on apps that can have multiple windows open and rely on localStorage for data.

Reproduction

• Create a new Tauri project using the latest Tauri versions
• Make a simple button to update a key in localStorage and one to print the current value of the key
• Run npm run tauri build and start two instances of the compiled binary (./src-tauri/target/release/(app-binary) in two terminal windows)
• Click the buttons in both windows and observe how localStorage is not synchronized across both instances. It doesn't sync until you close and reopen an instance.

Expected behavior

localStorage, sessionStorage, etc. should be synchronized across all instances of an app, especially when the windows have the same label!

Full tauri info output

[✔] Environment
    - OS: EndeavourOS Rolling Release x86_64 (X64) (KDE on wayland)
    ✔ webkit2gtk-4.1: 2.50.1
    ✔ rsvg2: 2.61.1
    ✔ rustc: 1.90.0 (1159e78c4 2025-09-14)
    ✔ cargo: 1.90.0 (840b83a10 2025-07-30)
    ✔ rustup: 1.28.2 (e4f3ad6f8 2025-04-28)
    ✔ Rust toolchain: stable-x86_64-unknown-linux-gnu (default)
    - node: 22.19.0
    - pnpm: 10.19.0
    - yarn: 1.22.22
    - npm: 11.6.2
    - bun: 1.3.1

[-] Packages
    - tauri 🦀: 2.9.1
    - tauri-build 🦀: 2.5.1
    - wry 🦀: 0.53.5
    - tao 🦀: 0.34.5
    - @tauri-apps/api  ⱼₛ: 2.9.0
    - @tauri-apps/cli  ⱼₛ: 2.9.1

[-] Plugins
    - tauri-plugin-sql 🦀: 2.3.0
    - @tauri-apps/plugin-sql  ⱼₛ: 2.3.0
    - tauri-plugin-shell 🦀: 2.3.1
    - @tauri-apps/plugin-shell  ⱼₛ: 2.3.1
    - tauri-plugin-log 🦀: 2.7.0
    - @tauri-apps/plugin-log  ⱼₛ: 2.7.0
    - tauri-plugin-clipboard-manager 🦀: 2.3.0
    - @tauri-apps/plugin-clipboard-manager  ⱼₛ: 2.3.0
    - tauri-plugin-fs 🦀: 2.4.2
    - @tauri-apps/plugin-fs  ⱼₛ: 2.4.2
    - tauri-plugin-dialog 🦀: 2.4.0
    - @tauri-apps/plugin-dialog  ⱼₛ: 2.4.0
    - tauri-plugin-opener 🦀: 2.5.0
    - @tauri-apps/plugin-opener  ⱼₛ: 2.5.0

[-] App
    - build-type: bundle
    - CSP: unset
    - frontendDist: ../dist
    - devUrl: http://localhost:1420/
    - framework: React
    - bundler: Rollup

Stack trace

Additional context

I'd like to draw attention back to #10981. The issue is closed but is not resolved!