Difficult session persistence scenario

Need help choosing the right persistence method for the following situation. We have two webservers with two IP addresses each. All 4 IP's listen on TCP 443, as there are two different web services applications hosted on each webserver, as shown below:

Webserver 1 IP-Addr-1 Port 443 AppA IP-Addr-2 Port 443 AppB

Webserver 2 IP-Addr-1 Port 443 AppA IP-Addr-2 Port 443 AppB

F5 VIP 1 Webserver1 IPAddr1 Port 443 AppA Webserver2 IPAddr1 Port 443 AppA

F5 VIP 2 Webserver1 IPAddr2 Port 443 AppB Webserver2 IPAddr2 Port 443 AppB

Need to figure out the best way to keep the same client IP, as it goes to access both Application A and Application B, on the same Webserver despite being accessed through different VIPs.

Webserver 1 and 2 do not share their internal session information with the opposite server; if a client first accesses Application A through Webserver 1, but later accesses Application B through Webserver 2, this will cause the application experience to break.