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I have a main document (manuscript) and a supplemental document.

I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized?

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The xr package is probably what you need.

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    And also xr-hyper if hyperref is needed.
    – egreg
    Apr 25 '12 at 14:31
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You can use the xr package with the following setup in the header of both files

\usepackage{xr-hyper} \usepackage{hyperref} \externaldocument[]{name_of_other_file_without_extension}

You can then just use regular \ref{fig:label name} like you would in the same document.

Placing this header info at the bottom on the header will likely reduce package conflicts regarding hyperref.

source for more reading.

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