I think it depends on the height of your run and its suitability to take a roof. Our big run is a converted pergola, so was easy to roof and once we did it, we wondered why we hadn't done it before. Our lot love it and rush back if even the slightest drop of rain falls when they are in the garden. They don't 'do' weather
Its not too bad if you only have a small coop and run set-up. If wooden, they tend to have a wire mesh roof supported by a wooden frame that is fairly easy to cover. Thinking about it, we roofed Alphie's old wooden coop and run set-up for him and his 2 girls with a couple of sheets of that opaque plastic roofing. It let diffused light in, but didn't get too hot, both of which are considerations for a low roofed run.
If you have a home built larger run, it could be a lot more difficult to cover, especially if it currently doesn't have a roof. A tarp would need a fair bit of support to stop water pooling in it and dragging it down, plus a lot of tethering in windy areas. It might be easier to put something up inside to reduce the run area and then cover it. It maybe worth trying to get hold of an old pop-up gazebo, tent frame or similar, covering the top, then netting off any gaps between it and the top of the run to stop wild birds getting in. A sort of run within a run, rather than worry about adapting an existing structure.
For free rangers, especially if they have a decent sized flock, this must all be a major headache.