All the nodes that have been online long enough will know that the fake chain is fake. Because it doesn't match the state that they have been observing all along.

So as long as you have no extended downtime, your nodes know what the right chain has to look like. Ethereum dynamically adds checkpoints so that block reorgs can't reach too far into the past but that is mostly a convenience function as it alone couldn't solve this issue.

Now, there is a problem if you are a new participant. Then you can't decide which chain is the right one. You could observe for a while and choose the one that has more validators but in theory, there's the possibility that there's a contentious hard fork going on.

In this case you need to get the information which chain is the right one from outside sources. E.g. like your preferred exchange or the official subreddit.

This is the most glaring difference between POS and POW. With POW, you don't need any outside information as you can just verify which chain needed more work to be created and this is by definition the correct one.

POW leverages physics to be completely self contained for the price of energy consumption.