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I'm trying to retroject Natural Earth's Shapefile of the world (Admin 0 – Countries on this page: https://www.naturalearthdata.com/downloads/10m-cultural-vectors/).

Here's the original Shapefile in QGIS 3.4.1:

enter image description here

Reprojected in Robinson World, centered at Long 0:

enter image description here

All seem good. However, when I try to reproject centred at a different longitude (say Montréal at -73.3), here's what happens:

enter image description here

The polygons aren't sliced at the edge of the new map, and the software is confused as to where polygons start and end.

What's a good way to deal with these problems?

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    You have to manually slice the polygons before you can re-project them centered around a different longitude. I couldn't find it, but I know this question has been asked on GIS SE before and has a helpful answer. Try looking for questions with the terms "meridian" and "projection".
    – csk
    Jan 17 '19 at 20:25
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    This one @csk ?
    – MrXsquared
    Jan 17 '19 at 20:48