Magnetic coordinates and lines drawing across the world map

I'm extending basemap to work with magnetic coordinates. Basically, latitude longitude are converted to/from magnetic coordinates during calls to Basemap.__call__(). However, I get results like this:

It seems that coastlines/polygons are not properly "broken" on the left/right plot edges. I suspect I have to change something in _readboundarydata() to fix this (since that's where the coastlines and polygons are initiated), and I have had a look at it, but I do not really see what I would need to change. I don't even see how this is handled in the normal (geographic coordinates) case. There are some references to the crossing of east/west datelines, are these "datelines" the left/right map boundaries in projections like the above? I don't really understand what this part of the code accomplishes.

Any tips?

Start by reproducing the magnetic-coordinate map shown in figure_1 and trace how Basemap.call() transforms coordinates before _readboundarydata() creates coastlines and polygons. Read the existing east/west dateline-crossing logic to determine how it relates to projection boundaries. Done means coastlines and polygons are correctly broken at the left and right plot edges for magnetic coordinates.