You actually have a good place to use white subway tiles! Select a 3x6 or a 4x4 tile that's not builder basic; something that has a nicer look to it, undulated surface for example. My preference is glossy, but a matte finish will also work.

I would suggest simple subway tile in a light gray found in the counter material. A white backsplash would jump out at you, visually. A light gray backsplash would simply blend in. Is there any need for privacy in this kitchen….a window shade or curtain on the window and door? Could the darker paneling above the backsplash area be replaced or painted to blend in with the backsplash?

If you like the copper idea, consider backpainting tempered glass with a copper paint. Consider simply installing metal flashing or corrugated metal to carry out your rustic theme.

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No white in your space. It just won't look good with the more rustic flavor you've got going, and hickory needs something with a bit of visual heft to balance it.

I have hickory cabinets and just had a matte dark green subway put in. No pics, the space isn't finished yet, but just want to throw it out there that with hickory you really, really need something that will visually set it off but not compete with it. The grey suggestion above is a good one as long as the color plays well with your floor (which your counter does). Maybe try some blue or green samples, something that ties in with the floor but isn't grey (I can't tell on my monitor whether the floor leans more blue-grey or hints of green in there or maybe both).

BTW: Your cabinets are beautiful, and you made a great choice in counter -- love it!

Hard to see color accurately onscreen, but I think I see various mossy greens in the slate. If you can find those sort of colors in a rusticated subway tile, I think it would look great. A close runner-up would be to match to the gray in the counters, as previously suggested.

Pale grey to keep your lovely color palette going!

I'd prefer a warm off-white to reflect the warmth of the hickory and to balance the cold/dark of the countertop & black appliances.

Thanks much for all the feedback! Sending me in a better direction. I think I’ll do a heavy clean and reseal on the slate before I make a final choice.