I do not believe that you can do this without rewriting the story. I just cloned the same repository twice to run the test. I made the same change in both repositories and then committed it with the same journal message ("foobar"), with the only difference being that it was signed and the other not.
parent 50c6dd65f1d7a240cf6b5c9585ce363ef4708d1e
new b3ff731922f80a417b84ed492537c1f7ba74715e
parent 50c6dd65f1d7a240cf6b5c9585ce363ef4708d1e
new 688b3be2e55558c45b00b6a6c02086a03768e02d
As you can see, starting with the same parent (50c6dd65), the result is two different commit hashes. Thus, for undefined commits, this is no different from any other rewriting of history (and therefore carries the same obligations).
In response to your comment asking if the hash has changed just because of the difference in timestamps, I don't believe that. If you check with cat-file:
$ git cat-file -p 688b3be2e55558c45b00b6a6c02086a03768e02d
tree 074e53e54670dea3502229e9494f3d571f5dcc16
parent 50c6dd65f1d7a240cf6b5c9585ce363ef4708d1e
author Dan Lowe <dan@XXXXXXXX.com> 1448768563 -0500
committer Dan Lowe <dan@XXXXXXXX.com> 1448768563 -0500
gpgsig -----BEGIN PGP SIGNATURE-----
Version: GnuPG v1
iQEVAwUAVlp0N1rGfrtJ2k+kAQIbYQf7BLx3/jqU/vwvoJOcbq5MPK0ok7B8mOaF
VWhNCbAeOBMzXdrn8IQxY2xYcPsy+d6pNx6ZOF70L3VZm6rWFxNzZQRrjS4ByOAM
VyoL8bXceMcrb/sQUHM5yTCaDcfoYx40K0q91XsGew2EIzNKcOraK1Ee4hEtPg1D
ojyPVjiWz2qMJJ0YYAATSvWwlKFO2ShTC6tGZDHrx0e6BAEN5QS4KdGhNech/vpU
IPFDjIKWtGPTbYY8Z95vKLAMYWPZDJ8j/x1gRytN8PDjRufRtpRnZMccB6JQoXNZ
5L23WQFfUFeXRdWf0MtkrbrSwzuaaIF8l1oGYnEtYT6nOIktPT47Fw==
=/U9b
-----END PGP SIGNATURE-----
foobar
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