More grouting! Lots and lots of grouting. I finally finished all the walls and had some left over grout already mixed up, so despite my mental thinking that I'd do the floor on a later day, I went ahead and did the floor. Wow, the floor is easy to do in comparison -- mostly because the grout that doesn't make it into the joints isn't constantly falling down on the ground.
To complicate things a bit, the natural stone floor tiles come with instructions to seal them before applying grout, to avoid staining them. The sealer says it needs to sit on for five minutes then wipes off, but of course says for "pre-grouting applications, apply twice" and you need to wait 48 hours between applications. I decided to ignore that, but did worry a bit about staining our floor. (Though admittedly staining white/gray/black tiles slightly whiter isn't really that noticeable.) I don't think they stained in the end though.