`Export` such drawing to `pdf` is not a vector format `pdf` file. We use `Mesh` to draw such hatchfilling instead.

```
plot = RegionPlot[Sin[x] Sin[y] > 1/4, {x, -10, 10}, {y, -10, 10}, 
  BoundaryStyle -> Thickness[0.01], PlotStyle -> None, Mesh -> 80, 
  MeshFunctions -> {#2 - #1 &}, 
  MeshStyle -> {RGBColor["#0077BB"], AbsoluteThickness[1.5]}]
Export["test.pdf", plot] // SystemOpen
```

[![enter image description here][1]][1]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/3VM5a.png
  [2]: https://i.sstatic.net/bvwPO.png