"Suppose Alice has hashgraph A and Bob hash hashgraph B. These hashgraphs may be slightly different at any given moment, but they will always be consistent. Consistent means that if A and B both contain event x, then they will both contain exactly the same set of ancestors for x, and will both contain exactly the same set of edges between those ancestors."

Consider UTXO-based events. There can be an event E1 that consumes UTXO1 and UTXO2 and event E2 that consumes UTXO2 and UTXO3. Hashgaphs that contain one of these events are consistent but their union is not. This can be used to perform some byzantine things, I can think of at least two of them: doublespend and degradation of service.

This paper is a clear example of how to make a thing that has no obvious problems.

I haven't read that paper but it seems like it's fixing a different problem of Byzantine fault tolerance. Most consensus systems that are internal for an organization don't have the Byzantine issue so it simplifies the problem.