Thanks Izzy for the tip, Open Camera indeed has a HDR mode which is real HDR. You can actually see the app take 3 pictures (under-exposed, normal, over-exposed) one after the other. In the settings you can even set the app to save all 3 in addition to the HDR result, if you want to.
Here is how to enable HDR mode:
In the result, you can see that the bottom of the picture is not too dark (compare with bottom left of the non-HDR screenshot above), despite the direct sun at the top of the picture:
Open Camera is free and open source.