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Of the few things that bother anemone, this looks like rust. Have you seen any insects? Flea beetle damage can look similar.

To manage rust, clip off all damaged leaves and discard with trash. Work in the plants only when the leaves are dry. Keep leaves dry when watering, or water early so leaves dry before nightfall. Plant debris should be removed and destroyed, not composted. Use fresh mulch under plants to help prevent soil from splashing onto leaves.

Stone-fruit rust (attacking cherries, peaches, plums, apricots, and almonds), is caused by a rust that spends part of its life cycle on anemones and liverleaf. A number of rusts may commonly be prevented by dusting or spraying plants with garden sulfur labeled for management of rust. Neem oil labeled for control is another organic treatment. Neither of these will cure an already infected leaf. Please read and follow all directions and cautions when using chemicals. If you decide to use a non-organic fungicide, there are several available in the garden center.

You can submit fresh samples of leaves both healthy and diseased, to MSU Plant Diagnostic lab. Print out a submittal form and mail to the lab. Don’t mail samples on Thursday or Friday or they will be too wilted after a weekend in the mail to be of any use. The lab can examine the samples under a microscope and give you a definite diagnosis, along with management options. There website- https://pestid.msu.edu

Here are some references for you- (rust can look different on different plants,so the pictures may not match anemone exactly)-

https://pestid.msu.edu/plant-diseases/rust-diseases-other/

http://ipm.ucanr.edu/PMG/r280101211.html