[FEATURE] Global CLAUDE.md instructions should persist across Claude Code for Cloud sessions (desktop + mobile)

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• [x] This is a single feature request (not multiple features)

Problem Statement

When using Claude Code for Cloud (the web-based version), there is no way to persist user-level instructions across sessions or devices. The ~/.claude/CLAUDE.md file is machine-local and does not apply to cloud sessions at all.
Problem:
A user working on Claude Code for Cloud on desktop has no way to carry their preferences or standing instructions to a mobile session — or even to the next desktop session. Every session starts completely fresh with no memory of past mistakes or established conventions.
Real example:
I had Claude set up a GitHub repo without first creating a main branch. This caused a cascade of manual fixes (default branch misconfiguration, failed GitHub Pages deployment, etc.). I wanted to add a rule so Claude would never do this again — but there is nowhere to put that rule that would survive across cloud sessions on different devices.
Request:
Add a cloud-synced user profile or global instruction set for Claude Code for Cloud, tied to the user’s Anthropic account, that:
∙ Applies to every cloud session regardless of device
∙ Works on both desktop and mobile app
∙ Functions like a global CLAUDE.md but stored server-side

Proposed Solution

No specific solution to suggest

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

Trying out Claude Code on mobile for the first time from the mobile app, I asked it to bootstrap a new GitHub repo. Claude pushed all work to a feature branch but never created main first. This caused a cascade of 5+ manual fixes — creating main, changing the default branch, fixing GitHub Pages environment rules, and re-triggering the failed deployment.
When I asked how to prevent this in future sessions, Claude suggested ~/.claude/CLAUDE.md — but that file doesn’t exist in cloud sessions and doesn’t sync across devices. There is no way to give Claude persistent instructions that carry over between Claude Code for Cloud sessions on desktop or mobile. Every session repeats the same mistakes from scratch.

Additional Context

_No response_