Any natural way to dye outdoor curtains?

You can use fabric dye

Cut designs from fabrics with blue colors or designs in it then stitch on curtains in whatever pattern you desire. Quilted look, maybe stripes, solid circles, or rings interlocking or not. Whatever your taste. This would give you option to use or include many shades of blue with or without other colors too. Add throw pillows of same colors or designs for a more finished look.

Have you Googled yet? How about this?

Shall I assume you used the 100% cotton dropcloths?

Check the internet for natural or organic fabric dyes if you want "natural".

You can transform a drop cloth with Rit dye

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