Guide
How the kit works
Start with Sword to Mark Tide's Trace when possible. It applies two Aero Erosion stacks and leaves a Sword of Discord's Shadow. Follow with Basic Attacks: the fourth hit of Sword to Carve My Forms applies one stack and leaves a Sword of Divinity's Shadow. A Heavy Attack can create the Discord shadow if you did not enter with the Intro Skill.
Use Sword to Bear Their Names for two more Aero Erosion stacks and a Sword of Virtue's Shadow. Then plunge from the air to recall the shadows. Each type grants a different effect for Manifest: Discord spreads stacks, Divinity strengthens Aero Erosion, and Virtue gives interruption resistance and lets an attack Stagnate enemies.
Cast A Knight's Heartfelt Prayers to enter Manifest as Fleurdelys for 12 seconds. With the Divinity effect, nearby enemies take 50% more Aero Erosion DMG and its damage interval is halved. Attacks from Tempest restore Conviction; Fleurdelys's two-part Resonance Skill also pulls nearby targets together.
At 120 Conviction, use Blade of Howling Squall (liberation). It ends Manifest, restores 50% of Max HP and removes Aero Erosion from targets hit. Each stack removed amplifies the damage they take from the finisher by 20%, up to five stacks.
Build and why
Use five pieces of Windward Pilgrimage. Its listed two-piece bonus gives 10% Aero DMG, which suits both Cartethyia and Fleurdelys.
Defier's Thorn is the first weapon choice. It gives Max HP, lets damage ignore some DEF after an Intro Skill or Basic Attack, and amplifies damage taken by a target with Aero Erosion. Red Spring comes next: it gives Basic Attack DMG Bonus, which also applies to Cartethyia's Heavy Attack and Resonance Skill because those hits count as Basic Attack DMG.
Blazing Brilliance is another option. It gives Resonance Skill DMG Bonus as it builds stacks, though Cartethyia's own Resonance Skill hit counts as Basic Attack DMG. Do not assume that every hit from a skill benefits from a bonus with the same name.
Leveling priority
I would level A Knight's Heartfelt Prayers first for the finisher, then Tempest for Fleurdelys's attacks. Follow with Sword to Carve My Forms, Sword to Bear Their Names, and finally Sword to Mark Tide's Trace. This is a suggested order, not a fixed rule.
For maximum ascension, gather 170,000 Shell Credit, 4 LF Tidal Residuum, 12 MF Tidal Residuum, 12 HF Tidal Residuum, 4 FF Tidal Residuum, 46 Unfading Glory and 60 Bamboo Iris.
Maxing all skills takes another 2,030,000 Shell Credit, 25 LF Tidal Residuum, 28 MF Tidal Residuum, 40 HF Tidal Residuum, 57 FF Tidal Residuum, 26 When Irises Bloom, 25 Inert Metallic Drip, 28 Reactive Metallic Drip, 55 Polarized Metallic Drip and 67 Heterized Metallic Drip.
Resonance Chain nodes worth it
My first pick is Freedom Found in Storm's Wake (sequence 6). The finisher no longer removes Aero Erosion and instead raises stacks on targets hit to their maximum. Targets also take 40% more damage from Fleurdelys.
I would consider Blade Broken by Tempest (sequence 2) next. Casting the Liberation raises the nearby Aero Erosion stack limit by three, and the next direct hit applies three stacks to targets in range. It also increases several Cartethyia attack multipliers.
Crown Destined by Fate (sequence 1) is another option. Fleurdelys gains 25% Crit. DMG at each Conviction threshold of 30, 60, 90 and 120, up to four stacks. The effect lasts 15 seconds without refreshing when new stacks are gained, and ends after the finisher.
Prisoner Hanged in the Tower (sequence 3) makes several Fleurdelys moves inflict two Aero Erosion stacks and increases the finisher's DMG Multiplier by 100%. Sacrifice Made for Salvation (sequence 4) gives the team a 20% DMG Bonus for all Attributes after an eligible status is applied. Hope Reshaped in Storms (sequence 5) helps with survival and reduces the Liberation's HP cost. If you are stopping before sequence 6, you can leave those later upgrades for another time.
Teams
Sanhua is a Glacio teammate option, and Ciaccona is an Aero option. Rover: Aero is also worth considering: A Heart's Truest Wishes gives Rover: Aero 25 Windstrings when they cast their Resonance Liberation.
Wind's Divine Blessing amplifies Aero DMG dealt by another active Resonator against targets with Negative Statuses by 17.5% for 20 seconds. An Aero teammate can make use of that effect. A healer is also useful alongside Cartethyia's HP cost; her passive increases healing received by other team members by 20%.
Common mistakes
Do not enter Manifest without recalling the shadows if you can collect them first. In particular, the Divinity shadow gives the Aero Erosion amplification and shorter damage interval.
Do not press the finisher as soon as it becomes available without checking Aero Erosion. It consumes the stacks on targets hit in the base kit, and up to five stacks increase its damage. Also watch Conviction: pressing Resonance Liberation below 120 transforms Fleurdelys back into Cartethyia instead.
Do not forget the second press of Fleurdelys's Resonance Skill. The skill goes on cooldown if you wait too long or switch to another Resonator before using the follow-up.