Clay siding is a dull, dun color. It's not lively, but the good news is that its very dullness makes it easy to find shutter colors that will go with it. When color is neutral, as in gray, black or white, it naturally matches any other color, as there is no possibility of clashing color. This leaves you free to pursue many possibilities for your shutters.
If you have existing trim, gutter or other colors, you can simply use the same color for the shutters as you have for the trim. The result is simple and harmonious. Whatever trim color you have will be fine with the clay siding, and matching the shutters to it can even be economical, if you have the original paint stored.
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Keep Things Simple
If you want a simple, classic match for the clay siding, stay with colors that don't shout too much. These will be colors that share the neutral qualities of the clay. For instance, a mocha color is a warm option. Olive is a cool gray-green that would pick up the gray of the clay but also provide some hue contrast. Terra cotta is a reddish tone that has a bit more intensity but is still mild. Go for a hue that is different enough in terms of darkness or lightness so the shutters do not get lost visual.
Step the contrast up a bit by selecting colors of more intensity. Goldenrod is a full, warm yellow-gold that would add spice to the exterior. Coral is another option that is rich with color but not too bold. Pea soup, orchid and teal are more varied hues that impart dash while not overwhelming the siding.
Try Something Different
Since the clay siding is so neutral, if you would like to be more audacious, you can select highly vibrant colors. If there are other colors present on the trim or doors, this may not be the choice for you, but if you want intensity, colors like fuchsia and chartreuse will do the trick. Colors like maroon and midnight blue add plenty of intensity but are less striking as they are darker in tone.