As long as the same ballot will be used not just for the runoff but also for the initial election, people have an incentive not to write down their true preferences (this is implied by the https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_... (approximately) which says that as long as there are >2 candidates, tactical voting is incentivized).

The advantage of actually going to the polls twice is that you eliminate tactical voting in the final runoff; people no longer have any incentive to lie about their preferences.

You get to honestly veto the candidate you dislike the most (or back the candidate you want if s/he actually makes the final ballot) in the runoff.

But if anything, the first vote becomes even more tactical potentially in bizarre ways such as aiming to ensure that a candidate sure to make the final ballot runs against the most odious fringe candidate on the final ballot.

Maybe, but I get the impression that overall the addition of a true runoff tends to make things better. For example, this guy compared a number of voting methods using his preferred metric, Bayesian regret; note that when a method and a variant with a runoff added to it are compared, the runoff tends to do slightly better: http://rangevoting.org/StratHonMix.html

In addition, I have no data for this but I expect that a runoff would increase the perception of legitimacy of an electoral result, because while the initial voting procedure that narrows the field to only 2 candidates will occasionally be impacted by counterintuitive results such as spoilers, at least the runoff can always be interpreted in a straightforward manner as "we were choosing between A and B, and most voters preferred A over B".

> Maybe, but I get the impression that overall the addition of a true runoff tends to make things better. For example, this guy compared a number of voting methods using his preferred metric, Bayesian regret; note that when a method and a variant with a runoff added to it are compared, the runoff tends to do slightly better

I suspect that result is caused by voters being misinformed about how IRV works. It would be interesting to see IRV compared to Exhaustive Ballot, which is like the Two-Round system, but with up to n-1 rounds for n candidates. The last-placed candidate is eliminated each round until there are a majority of votes for one candidate, just like IRV.

In practice it rather works the other way: the fact that your ballot will also be used for the final runoff reduces the incentive to tactically vote, because there's a greater chance your vote will backfire.