The easiest method would be downward "skirts" as mentioned above. A little more complicated one I'm thinking about is reconciling error maps on the border when an adjacent tile loads so that they split at the same point, and updating the corresponding mesh — should be fast enough to do on tile loads.

I think I've seen a rather pragmatic hack that has been used in Google Earth for their stitching of terrain: They simply add a kind of skirt at the boundary of each square chunk [0], so that there aren't any white/black pixels. There may be some artifacts between LoDs but they soon disappear when the next chunk gets more refined.

Edit: I think the Google terrain skirts were slanted, rather than falling straight down, so two neighbouring skirts intersect in a V.

[0] https://www.researchgate.net/figure/A-terrain-chunk-a-withou...