Can we assume that you are exporting PDF as opposed to creating PDF via distillation of PostScript?
Do you see the quotes when you open the resultant PDF file in Reader or Acrobat?
If you use Ctrl-D / Cmd-D to get the PDF file Document Properties, does the Fonts tab show that all the fonts are at least subset-embedded?
If all the above are “yes,” what is your supervisor using to view the PDF file with? Anything other than Adobe Reader or Adobe Acrobat and all bets are off.