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Combine geom_tile() and facet_grid/facet_wrap and remove space between tiles (ggplot2)

I have a data set with x, y, and z (resp) values along with two columns for facetting in order to create a grid of tile plots.

The output of dput() is at the end of the post.

library(ggplot2)
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My actual data set has six unique interactions between variables, however I've limited it to four for simplicity. I can create a tile plot with each unique combination, but not the whole set. The output of dput() is at …

r facet tile ggplot2

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facet ×1

ggplot2 ×1

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tile ×1