in which scenario is a full proxy TCP connection required?

IP Anycast is a network addressing and routing methodology where multiple nodes share an IP address and traffic is routed to the topologically nearest one - it is independent of TCP proxy mode.

Routing can be enabled on a device regardless of its proxy mode and does not require a full proxy TCP architecture to function.

In full proxy mode, the device fully terminates the inbound client TCP connection and establishes a separate, independently managed TCP connection to the server, allowing the proxy to pool, queue, and reuse server-side connections regardless of client connection state. This decoupling is the mechanism that enables server-side connection optimization, reducing the raw number of simultaneous TCP sessions hitting backend nodes. On F5 BIG-IP, Standard virtual servers operate in full proxy mode specifically to enable this kind of TCP connection management and optimization.

Full proxy mode specifically does NOT forward the original client TCP connection options to the server - it creates a brand-new independent connection; a half proxy or pass-through mode would be required to preserve and relay client-side TCP options.