I expect you are looking at the exported JPEG photos with some program which does not properly take the color space into account — it just assumes sRGB, the standard default. So, only your photos exported to match that expectation work. (Even if your screen is calibrated, your applications might not be so smart.)

Keep in mind that Adobe RGB is a color space mainly optimized for print. While monitors are getting better (and yours may qualify as better), most consumer-grade ones can't cover the whole gamut anyway — there are colors that just can't be shown, so they're "clipped" to colors within the monitors' natural/native color gamut.

This isn't part of your question, but:

ProPhoto is a very large color space designed to be a working space, not used for final results. It's very big, and in fact not only exceeds what your monitor can show but contains "imaginary" colors (that's the technical term) which don't correspond to anything in human vision. That's good for a working space, because you don't accidentally clip colors in the midst of curves adjustments and etc., but it's not very useful for a final format. In fact, it's particularly poor for 8-bit-per-channel JPEG images, because there's only 256 possible values for each channel, and when those are spread out so far and "wasted" on imaginary colors, you have less data for real colors, possibly making posterization an issue in gradients.

So, back to the question: you should save your images in a color space based on the intended display. In the current world, that's largely sRGB, unless you're making your own prints or working with a lab that provides printer-specific color profiles for your use. Adobe RGB may also be an option, but your viewing/presentation choices will be more limited (and I think that's what you're running into here).