So if there is no proof of work how do you avoid forks? The author says something about splits being detectable but that doesn't really help us decide on which side of the split is correct.
Forks are ok as long as they can be merged in the short term.
If something forks for a long time and stays forked, there is hardly any reason to establish a total order during the merge!
Think of an IRC netsplit for example. One that happens for a few seconds may attempt to merge back the chats in some fair order they were made, in diff forks.
But if the netsplit happens for a whole day, or month, no one really gives a crap about ordering messages across forks. The merge is too complex! In fact, the resulting conversation would be MORE nonsensical than if you correctly rendered the split conversations as a DAG in the client.
Similarly, if bitcoin forks into bitcoin cash or whatever, and enough validators accept it, I get to "double-spend" my new money now. Proof-of-work is no panacea. If we religiously want consensus then no transaction can ever be truly confirmed - there is always a chance some larger fork comes along and undoes all transactions on my fork going back a whole month. Interplanetary File System has to deal with this.
The problem is that we still haven't evolved our thinking about currencies as DAGs and keep worrying about the double spend problem and turn to global consensus to fix it.