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I have reviewed all past recommendations around spell checking, language and proofing and cannot find an answer that resolves this issue.
That is where the problem lies, though, and its solution. Read through Mastering the Spelling Checker. It is a fairly long article but has the solution to your problem.
Try selecting your entire document body (Ctrl+A) and check the language setting. Is the checkbox for "Do not check spelling or grammar" empty or does it have a check mark or does it have some sort of color? If it is anything other than empty, check the box
until it is empty. This is a character-level setting, not by paragraph, table, or document.



If this doesn't work, send me a sample document with your tables (clear of data if you wish) and I'll look at it. Do not send it to me, though, before you have read the entire article.