Two repairs
The Hellion's Heart is the quest for Karlach's engine. Dammon does the work in two stops.
The first Dammon upgrade in Act 1 still leaves Karlach too hot to touch properly. At Last Light Inn in Act 2, Dammon can repair the engine. The partner scene is available on a later long rest after both upgrades.
Hot Date, the achievement for a date with Karlach, is listed on the romance overview.
The long rest after Last Light
After The Chosen of Shar, the partner scene can no longer start. Leaving Act 2 before that scene ends the romance when Act 3 starts.
The romance page marks an Act 1 night-visit scene as bugged, and treats the grove celebration as the reliable place to start the romance.
Keep the path to Dammon open
He has to be there for both conversations. Finish the Act 1 upgrade before a grove-resolving exit, and keep him alive through Last Light for the second one.
Save the Refugees ends on either Act 1 exit. If Kagha's quest is done and the goblin leaders are not, the tieflings are gone. Karlach leaves if the paladins of Tyr from Hunt the Devil are still alive when you take them.
Two pieces of Infernal Iron
Both engine upgrades together take two pieces of Infernal Iron. Karlach has to be in the party, because she installs each upgrade herself. After the first she is still too hot to touch. After the second she can touch people again.
If the first upgrade is in the grove, Dammon does the second at Last Light. If the first meeting with him is at Last Light, both upgrades can be done after a long rest between them.
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