In the second measure here (source) two notes marked with stacatto dots are also tied. Aren't these two things incompatible? How is this to be understood?
It's a red herring! It's not a tie, and they're not staccato, per se. It's a separate sign called 'portato',or more accurately and easily understood 'articulated legato', and if it was applied to notes that were not the same, obviously it couldn't be a tie. A slur it would be. Now, you can see that two slurred notes separated because they need to be shortened is how it's played. Usually for strings, as in the same bow stroke, with a small gap between the playing of each note. Brass players also find portato, and they would probably tongue the second note gently, to produce separation without much attack on the second note. I guess that the second 'staccato' sign actually has no effect on the second note - it's just part of the potato sign that needs to be there.