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It might be possible but I wouldn't suspect so. When the PDF opens it will show the print dialog; if you are able to redirect in the first place (using embedded Javascript in the PDF) then the user might lose the ability to print (since the document isn't there anymore).
"Printing" in PHP means outputting to the browser, not printing like to a user's printer. Getting the PDF as a string just means that you don't have to use a file (on the server) to hold the contents, which affords you a bit more flexibility in terms of what you do with it. However that doesn't impact the ability to have it printed for the user.