The simple reason why QGIS can not draw these polygons is because some of the vertices are on the backside of the globe, and QGIS can not draw a closed polygon with the rest. The GDAL ogr2ogr function in the background is programmed to kick off the whole feature when such an error occurs.

So you have two solutions:


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**Solution 1: Clip your polygons to the hemisphere**

You need `Numerical Digitize` and `CAD Tools` plugin for that.

 - Set the project CRS to ortho projection
 - create a new point layer in the same projection
 - with the Numerical Digitize plugin, create the follwoing points:

>     x y
>     6370000 0
>     0 6370000
>     -6370000 0
>     0 6370000

(should also work with `Add delimited text layer`)

 - enable snapping on the point layer
 - with CAD Tools, select the first three points and create an arc from them
 - do the same for point 2 to 4 for the southern hemisphere
 - change project CRS to WGS84
 - Save the CAD layer as WGS84 and add that to the canvas
 - Save your boundary layer also as WGS84 and add it to the canvas
 - convert the CAD lines to a new polygon layer
 - remove all except the polygon layers
 - clip the world boundaries to the CAD polygon layer
 - set project CRS back to ortho

You should get this picture:
![world vector in ortho projection][1]

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**Solution 2: Convert your vector data to raster**

 - in Settings, disable `on-the-fly-reprojection`
 - Save your boundary layer also as WGS84 and add that to the canvas
 - Use `Raster -> Convert -> Rasterize`
select x=360 and y=170 as dimensions for the raster (or n*360 and n*170)
 - Use `Raster -> Projection -> Reproject` to the ortho projection under another name and add it to the canvas. You will get some errors, but the rest of the raster will do. Don't bother that you see a grey rectangle
 - Rightclick on the raster layer -> Properties, Style tab
 - choose pseudo colours
 - in transparency tab, choose 0 for 100% transparency
 - Remove the first raster
 - set project CRS to ortho

you should see the follwing picture:
![enter image description here][2]

The same method works with rasters, as you can see in my avatar picture ;-)

  [1]: https://i.sstatic.net/nCGES.png
  [2]: https://i.sstatic.net/SA5K7.png