[FEATURE] Always-visible tab bar for quick switching between Chat, Cowork, and Code

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

The Chat / Cowork / Code tabs are not always visible in the desktop app. Switching between modes requires extra navigation steps, which interrupts workflow — especially when moving frequently between Chat and Code or Cowork.

Proposed Solution

Keep the Chat / Cowork / Code tab bar persistently visible at the top of the app, regardless of the current mode or screen state. One click should always be enough to switch modes from anywhere in the app.
Why it matters
Quick mode-switching is a core part of daily usage. Having to hunt for the tab bar adds unnecessary friction. A persistent tab bar would make the app feel significantly more responsive and efficient.

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

Related
See also #40750 (proposes unifying the three tabs into one adaptive interface — this request instead asks to keep the tabs but make them always accessible).