A tweak to IRV to make it a Condorcet method

tl;dr: Instead of eliminating the candidate in last place, eliminate the loser of a head-to-head matchup between the bottom two candidates. This ensures the Condorcet winner (if one exists) is never eliminated.

Background: how IRV works

In instant-runoff voting (IRV), each voter ranks the candidates in order of preference. The winner is then determined by repeated elimination, using the following procedure:

Let X be the candidate who is the first preference (among remaining candidates) on the fewest ballots, and eliminate X.
If only one candidate remains, elect that candidate. Otherwise, return to step 1.

A major drawback of IRV is that it is not a Condorcet method. That is, IRV can elect a candidate A even when another candidate B would beat every other candidate (including A) in a head-to-head matchup.

The tweak

We can tweak IRV to make it a Condorcet method by changing what happens in step 1 of the above procedure. Rather than automatically eliminating the last-place candidate, we hold a head-to-head matchup between the bottom two candidates and eliminate whichever one loses. The tweaked procedure is as follows:

Rank the remaining candidates according to the number of ballots on which they are the first preference (among remaining candidates). Let X be the candidate in last place in this ranking, and let Y be the candidate in second-to-last place. If there are more voters who prefer X over Y than voters who prefer Y over X, then eliminate Y; otherwise, eliminate X.
If only one candidate remains, elect that candidate. Otherwise, return to step 1.

Why does this make it a Condorcet method? A Condorcet winner, by definition, beats every other candidate head-to-head. So the Condorcet winner (if one exists) can never lose one of these bottom-two matchups, and therefore can never be eliminated, and therefore must be the last candidate standing.

Ties: In both standard IRV and tweaked IRV, there may be ties when two candidates appear as first preference (among remaining candidates) on the same number of ballots. These ties can be resolved in tweaked IRV using the same methods as for standard IRV.

Example

Percentage of voters	Ballot ranking
38%	(1) Alice, (2) Bob, (3) Charles
15%	(1) Bob, (2) Alice, (3) Charles
15%	(1) Bob, (2) Charles, (3) Alice
32%	(1) Charles, (2) Bob, (3) Alice

Bob has the fewest first-place votes (15% + 15% = 30%) but is the Condorcet winner, so the two methods diverge immediately:

Standard IRV: Bob is eliminated in the first round.
Tweaked IRV: Bob (last place) faces Charles (second-to-last place) head-to-head. Bob is preferred over Charles by 38% + 15% + 15% = 68% of voters, so Charles is eliminated instead.

This tweaked version of IRV is also known as "Bottom-Two-Runoff IRV".
More info: https://electowiki.org/wiki/Bottom-Two-Runoff_IRV