Ways to correlate client side and server side connections

if you're talking about active connections, this information is usually stored in the connection table. if you run the: tmsh show sys conn cs-client-addr <client ip> cs-client-port <client port>
command , you'll be able to see all active connection for the given client address, and the correlated server-side connection. 

if you want to filter for a specific Virtual Server, you can add the: cs-server-addr <VS ip> cs-server-port <VS port> 
options, and the output will only match active connections on selected VS for the selected client, one row for each connection. 

if more details is required, you can also append the "all-properties" option to the command in order to see informations such as input/output throughputs, the configured & residual idle timeouts, or lasthop mac addresses. once again - one entry per active connection.