[[FEATURE] Remote Control should preserve conversation context

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

When connecting to a running Claude Code session via Remote Control from another device (e.g., phone), the connection starts with a blank context. No prior conversation history is available.

This makes Remote Control functionally equivalent to starting a fresh claude session (not even claude -c). The only thing it shares is the working directory.

Proposed Solution

Remote Control should allow the connecting device to continue the existing conversation with full context, seeing prior messages and being able to pick up where the session left off. The remote client should receive the conversation history from the host session upon connection.

Alternative Solutions

Currently I SSH into my machine from my phone to interact with the same terminal session directly, which preserves full context since it's the same process.

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

1. I'm working on a Rust project on my MacBook with Claude Code
2. I step away from my desk and want to continue from my phone
3. I connect via Remote Control
4. Claude has no idea what we were just working on
5. I have to re-explain the context or just SSH instead
6. With this feature, I could seamlessly continue the conversation from my phone

Additional Context

_No response_