I have to say the Chris, despite the pinpoint sharpness of the orchids, the background does not look natural.
That in my experience is a common problem—
attribute might be a more accurate term—when shooting close-ups for a focus stack. You need a relatively wide aperture to help the stacking software get the entire subject perfectly focused, but that means the background is likely to be completely blurred, and you want to avoid a very busy bokeh that might detract from the overall image.
In the case of the orchid picture above, because of the way the buds were growing I was pointing the camera down at a potted plant that was sitting on a light-colored floor mat with a very fine weave. I might have been able to darken the background in post, but I doubt that would have made it look any more natural and I think reducing the contrast between foreground and background would have been a bad idea.
Below is another example of a floral focus stack with an "unnatural" background—in this instance a macro shot of a tiny azalea flower about 3 cm in diameter. I shot the subject against a black background because I figured that was the best way to bring out the range of colors and textures.

Finally, another orchid, but this one has a natural-looking background. Again, it was a potted plant but because I was shooting it with a level camera, I was able to position it outside a window with a garden view. There's just enough detail in the background to give the impression it was made outdoors in a single frame, rather than in a multi-exposure focus stack.

As you can probably guess, I'm still learning how to exploit this technique for close-ups, and feeling my way as I go along.