simplify tiled geometries while preserving anything on the edge

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I have a dataset of polygons split into evenly sized tiles. How can I go about simplifying them, but not along the tile edges, so that there aren't any gaps between continuous features when the tiles are aligned with one another? For instance, here are two unsimplified tiles:



After simplification, a "seam" forms between the two halves of Antarctica:



Here are Gists of the non-simplified and simplified GeoJSON files. I'm currently splitting the original geometries into tiles and then simplifying, and one solution appears to be reversing that order (simplifying and tiling afterwards), but that's unfortunately not feasible for architectural reasons. A PostGIS solution is preferred, but alternatives are welcome!



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