I'm confused. Is there some nice way to arbitrage this, or is this an inherent weakness of prediction markets and we should only trust positions they pick as leaders, not chances of their long tail?

It's a weakness, but not inherent. The bid/ask spreads are wide enough to prevent you from making any Dutch book bet, no matter how much someone can back you.

As more people join prediction markets the bid/ask spreads will shrink, but so will the cases of inconsistent probabilities.