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The supplementals-only query above gives me urls that also show in the regular search results without a supplemental tag. They even have the same cache date in both incarnations. In fact, some other site: operator queries also show the same url/cache date without the supplemental tag.
So what action can I sanely take on this basis? My concern is that these queries are tricking the site: operator into a worse-than-usual level of accuracy by asking them to perform outside the way they were designed.
This is completely normal. The Supplemental results are usually for words from an earlier version of the page.
See [webmasterworld.com...] and related threads.