Twins enter a room from different doors. Each gets a box with a button, and each time they press it, either a red or a green light flashes. If the twins are perfectly identical, you'd expect their lights to follow the same internal rules. But what if the rules differ even though the twins are exact copies?
That's precisely the scenario the authors explored. They took two identical quantum particles and performed measurements that don't interfere with each other (physicists call these commutative measurements). It turns out that the joint probabilities of outcomes, how often the lights flash in tandem, can sometimes be explained by a common hidden cause. But that cause either makes no distinction between the particles or contains nothing beyond the measurement outcomes themselves.
This leads to a strange conclusion. If the common cause is symmetric and doesn't distinguish the particles, it can't explain why their outcomes sometimes differ. If it does distinguish them, then the particles aren't truly identical: they have a hidden difference. So either the common cause is powerless, or the identical nature is an illusion. Physicists call this a dilemma: a symmetric common cause is either unnecessary or trivial.
Quantum indistinguishability can be compared to two drops of water from the same glass: even if you try to label them, nature itself forbids it — any label would change the state of the drop.
In Stanisław Lem's story 'The Seventh Voyage', twin heroes confuse everyone with their identical appearance, but in the quantum world, such confusion is built into the laws of nature.