In the current implementation, user interaction is necessary for clipboard access. When you programmatically triggering the paste, is this under an event (such as click, touchstart, mousedown, etc.)? If so, on Safari 13.1 and later, a "Paste" context menu action should show up, prompting the user to proceed with the paste.
The newer async clipboard APIs in WebKit work the same way — you can read more about it here: https://webkit.org/blog/10855/async-clipboard-api/
whsieh
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Is this on iOS, iPadOS, or macOS? If this affects iPhone and iPad, it sounds like your content may be laying out at desktop width (980px) and scaling down. I would recommend making your web content responsive by adding a meta viewport to mitigate this scaling:
<meta name="viewport" content="width=device-width, initial-scale=1">
You could consider overriding -[WKWebView performDragOperation:] for this case to perform your custom handling. Unfortunately, I don’t think .eml files dragged from Mail are sensibly exposed through drop events (i.e. via DataTransfer).