Apply a Filter to a Background Image

You can apply a filter to an entire element quite easily with the filter property. But what if you want to apply a filter just to the background of an element? It's weirdly tricky.

There are CSS properties that specific to backgrounds, like background-blend-mode — but blending and filters are not the same thing. It sorta seems to be the reason we have backdrop-filter, but not quite. That does filtering as the background interacts with what is behind the element.

There is no background-filter, unfortunately. What are we to do?