Turn Their Fleets to Dust

Destroy the last of a player's non-fighter ships in the active system during the space cannon offense step.

Secret objective – Action Phase

Action Phase

Destroy the last of a player's non-fighter ships in the active system during the space cannon offense step.

Source: Twilight's Fall

ClarificationsSelect rulings confidence level

tirules.com

If a player still has fighters in a system after assigning all hits produced during the Space Cannon Offense step, Turn Their Fleets to Dust may be scored.
If a player’s last ship in a system is destroyed, but an ability allows that player to then place ships into that system, Turn Their Fleets to Dust may still be scored.
Scoring Turn Their Fleets to Dust will not prevent that player from scoring a secret objective in the Space Combat step during this tactical action.
If a system contains only fighters, then Turn Their Fleets to Dust cannot be scored by destroying them.
Any effect that destroys non-fighter ships during the Space Cannon Offense step may be used to score Turn Their Fleets to Dust, not just hits produced by a space cannon roll.

Sources