What is required for a virtual server to support clients whose traffic arrives on the internal VLAN and pool members whose traffic arrives on the external VLAN.

This configuration is fully supported by BIG-IP; having clients on one VLAN and pool members on a different VLAN is a standard and valid deployment topology.

Enabling the virtual server on both VLANs is not required - the virtual server only needs to be present on the VLAN where client traffic arrives, not on the pool member VLAN.

A virtual server listens for incoming client connections on the VLAN or VLANs it is enabled for, so enabling it on the internal VLAN allows it to receive the client traffic that arrives there. Return traffic from pool members on the external VLAN is handled by the BIG-IP self IP addresses and routing configuration, which are separate from the virtual server's VLAN setting. Enabling the virtual server only on the client-facing internal VLAN is the correct and sufficient requirement for this topology.

Enabling the virtual server only on the external VLAN would make it listen for traffic on the pool member side, not the client side, so client connections arriving on the internal VLAN would not be accepted.