Great discussion here, everyone. Thank you so much! I'm sort of relieved that many others struggle with this. Actually when I was going through finds I had a hard time finding examples...that made me think, oh, maybe other people avoid this sort of garment too! (And then one of the ones I picked wasn't even two-color, good catch, FashIntern!)
I've been trying to figure out if these sorts of prints just don't suit me, or if there is something I'm missing about styling them. Reading through this discussion, I think one of the reasons that two-color prints are difficult for me personally is that they tend to have a very defined, graphic effect. And that's not a good fit for my style. (Again, for reasons I don't totally understand, Breton stripes are the exception to this rule.)
So, in thinking about how to work with them I need to find ways to tone down that graphic effect.
*Along those lines, I really like the idea of adding a tonal variation on one of the colors in the print.
*I like strong color contrasts, so the idea of using a complementary color is initially appealing. But in practice, I think this often enhances the graphic/color-blocky effect to my eye. (Again, that might be just the thing for some people! But it feels wrong for my physicality and/or style.) However, I think I could mitigate this by adding a contrast in value/tone/hue. For example, if I had a sky blue and white dress, don't pair it with peach (which I think would be my first impulse), pair it with rust.
*Similarly, the idea of matchy-with-each-other (but not with the print) accessories seems really logical -- but in practice again it often contributes to that too-graphic feel. So instead, finishing the outfit with mixed neutrals might be a better strategy for me.
*I do like the idea of pattern mixing...probably in my closet that means combining with Breton stripes, lol!
I do think that I'll probably have to accept that two-color prints are likely to be less versatile for me than multi-color ones. And maybe that's ok! But this gives me some great strategies to start with in thinking through individual items. Thank you so much, all!