A Some What One Arm Configuration on LTM

kend 

As mentioned by  Kevin_Stewart , one of the way is to use X-Forwarded-For HTTP header, so F5 will insert true client IP in the header and pass it to the backend server. With this, you will see requests from F5 IP only but in the header, you will find true client IP.

Second option which is not possible in most of internet facing applications. But if you have specific list of client source IP addresses who are going to access your site from internet, then on the server, you can put those IPs routes pointing to F5 and rest default route will go to core switch as you want. I know this is not an option for all the sites but just thought of it so shared. Ty!

Hope it helps!