If you lay it out the way I did you should be OK.The statement could
typically be simplified or reduced in various ways, e.g. drop the quotes
around the MERGEFIELD fields, especially if the MERGEFIELD rresults are
always numeric and are always represented the same way, e.g. "2" is always
"2" and not "02"

{IF "{MERGEFIELD SigBarNo}" <> "{MERGEFIELD A1Bar}"
"{IF "{MERGEFIELD SigBarNo}" <> "{MERGEFIELD A2Bar}"
"{IF "{MERGEFIELD SigBarNo}" <> "{MERGEFIELD A3Bar}"
"{MERGEFIELD Signatory}" ""}" ""}" ""}

should also be OK.

What hapens if you start with a really simple expression, e.g.

{ IF "{ MERGEFIELD SigBarNo }" <> "{ MERGEFIELD A1Bar }" "{ MERGEFIELD
Signatory }" "" }

or

{ IF "{ MERGEFIELD SigBarNo }" = "{ MERGEFIELD A1Bar }" "" "{ MERGEFIELD
Signatory }" }

? Does either of them work as you expect? If not, can you use Alt-F9 to
reveal the field codes, select each of the inner mergefield fields in turn,
and use the right-click menu to toggle field codes? You should then be able
to see the inner results and make sure they are as you expect.

Do you have any SKIPIF fields in your mail merge main document?