I seem have found the culprit, my single coated 67mm Hama UV filter.

I put it on as some point between these two pictures and didn't think it could be the cause because I have never had issues with filters before with any lens.

Here are before and after pictures with and without the filter, both at 400mm f/6.3 1/400 ISO250 on my 70D.

And 100% crop comparison:

EDIT: Since comments pointed it out, I did move a little bit to the left between those shots, was using autofocus, and was handholding my camera. This doesn't make for the most rigorous comparison but focus can't be blamed since we clearly see a difference in the bokeh which are of course both out of focus by definition. The "doubled image" effect isn't flagrant at the plane of focus, we can mostly see a more blurry image, but we clealy see some diagonal lines in the bokeh that are cause by this "doubled image".

Here is another comparison, this time mounted on a tripod and without changing the focus between shots, only removing the filter:

I am relieved that my lens is not faulty but am surprised of the quality deterioration that this filter is bringing. I have often heard that cheap filters could degrade IQ, but I had never seen it before with any of my lenses. I will be returning this filter, hoping it was a faulty one, but I found funny to find this in the product description:

Impact on the pictures: Clear, sharp photos Reduces haze and prevents fuzziness