Cats. The tileable image is in the attachment below.
To make images like this one, krita offers the wrap around mode (in the main menu, under "view". This shows the image tiled and you can paint freely on it without worrying about the borders.
What is also possible without ruining the tileability is applying filters that manipulate colours or values (like curves or the desaturation filter).
Other filters can ruin it (the image below show the result of G'mic bilater filter and a problem caused by it).
In this case the problem can be resolved manually painting over the borders.
Alternately (or in addiction), assuming that you are working with only 1 layer, this other approach can be quicker if the filter doesn't change the image radically:
- Duplicate that layer.
- Offset the top layer by its width/2 and height/2 (main menu->layer->transform->offset)
- Apply the same filter to both layers.
- Offset the top layer again by width/2 and height/2
Now you have two similar layers. When tiled, the bottom one has problems near the borders and the top one has problems in the midsections. So erase the midsections of the top layer with a soft brush and that's it.
Symmetry and rotations.
If you want a tiling with rotations or symmetries, the wrap around mode can be used in conjunction with the symmetry mode or the multibrush tool.
When the tiles are rotated 90° or 180° it's pretty straightforward but with 120° or 60° rotations, layer translations and the help of triangular or hexagonal grids are basically a necessity.