NATd traffic from internet not being load balanced

What kind of machine are you making the test request from or what program are you using? I may be wrong, but in looking at the tcpdump, I don't see client completing the 3-way handshake. Should be syn, syn-ack, ack The first two show the syn and syn-ack from BIG-IP, but then the client sends another syn instead of the ACK.

You should see something like this:

17:51:06.935331 IP 192.168.1.145.53926 > 173.59.8.62.443: Flags [S], seq 2008116475, win 65535, options [mss 1460,nop,wscale 5,nop,nop,TS val 1637511503 ecr 0,sackOK,eol], length 0
17:51:07.015095 IP 173.59.8.62.443 > 192.168.1.145.53925: Flags [S.], seq 1287488537, ack 3747984218, win 17896, options [mss 1388,sackOK,TS val 307343764 ecr 1637511503,nop,wscale 4], length 0
17:51:07.015142 IP 192.168.1.145.53925 > 173.59.8.62.443: Flags [.], ack 1, win 4128, options [nop,nop,TS val 1637511581 ecr 307343764], length 0

I'm assuming that the monitor is marking the server as Green (available).

So, you currently have the VIP set up with type "Standard" , can you try changing it to Performance Layer 4 and trying?

Try setting up FastL4 and that way BIG-IP won't do any TCP handling and you can see what a tcpdump of that looks like.

You may want to dump tcpdump to a file with "-w" flag to see more information and look at it in wireshark.