IRV Prime
IRV Prime is a voting method variation of instant-runoff voting (IRV) developed by Marcos Boyington that selects a single winner using votes that express each voter's order of preference, like IRV. It is a generalization of Condorcet IRV that works when there is no Condorcet winner.
Procedure
Find the instant-runoff voting (IRV) winners using the classic/standard rounds, where the candidate with least votes is eliminated & their votes redistributed (call this set Winners)
Find the candidates which satisfy the Scwhwartz set
Re-run the IRV election for each candidate in the Schwartz set, preserving that candidate as well as all Winners; preserve any candidate in the Schwartz set which defeats one of the Winners into Winners-Prime
Re-run the IRV election preserving all Winners and all Winners-Prime, proceeding with normal elimination when they are all that remain
Proof of satisfying Condorcet criterion as well later-no-harm
As this method simplifies to Condorcet IRV when there is a Condorcet winner, see proof on that method for how it's satisfied.
Because all candidates from classic IRV are preserved & eliminated in the final rounds, we know later-no-harm is preserved because early-eliminated candidates were guaranteed not to win.