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No, there's no way to forbid anonymous posting on the site. But I'll take a look at the image you linked to see if there's anything actionable there.
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[@ᏰᎯᎠᎻᎬᎯᏒᎿ":](/forums/meta/topics/ask-the-mods-important-questions-v2?post_id=4958824#post_4958824
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Just please don't link to leaks of personal messages on this site, ok? We can't control what people do on other sites, but on this site sharing the contents of personal messages without permission from everyone involved is not ok.