Does phasing out a permanent cause its auras and equipment to also phase out?

When a permanent phases out, any Auras, Equipment, or Fortifications that are attached to it also phase out at the same time. This is called indirect phasing out, as described in CR 702.26d. The attached permanents do not phase out on their own initiative — they are carried along by the permanent they are attached to.

Crucially, permanents that phase out indirectly do so simultaneously with the permanent they are attached to, and they phase back in at the same time as well (CR 702.26e). This means if you control a creature with a Aura attached to it and the creature phases out, the Aura disappears with it and reappears with it on the same phase-in step.

Also important: an Aura or Equipment that phases out indirectly does not trigger any 'leaves the battlefield' or 'enters the battlefield' effects, because phasing does not use the zone-change rules — the permanent is simply treated as though it doesn't exist while phased out (CR 702.26b).

There is one exception to note: if an Aura or Equipment phases out on its own (directly), it does not cause its host permanent to phase out with it. Indirect phasing only flows from the host to attached permanents, not the other way around (CR 702.26d).

Example: You control a Grizzly Bears equipped with Sword of Fire and Ice. An opponent's spell causes Grizzly Bears to phase out. The Sword also phases out at the same moment. On your next untap step, both the Bears and the Sword phase back in together — the Sword is still attached, and no enter-the-battlefield triggers occur for either permanent.