I don't think it's the game client anymore. Seems like my ISP is for some reason sending me packets out of order.

I get a bunch of these in my router's log:

Blocked incoming TCP SynAck packet from <remote ip>:<remote port> to <my ip>:<my port> with unexpected sequence

It only happens when another computer on the network is attempting to use a program that uses the same ports as FFXIV. The computer in question is accessing Facebook games... namely Zynga games. It's just weird to me that it causes my FFXIV session to terminate.