Despite what the tutorial says, Astra attacks reset the chain.
I used comet storm as the last move of a turn and it got countered by a revenge dummy. Somehow this made the chain counter display wrong for the next turn.
I exited out of the companion menu and my companion's face was still on screen, on top of everything else. Entering and exiting the companion menu again fixed it. I couldn't replicate it and I forgot to take a screenshot.