Attachments in signed messages don't show up in Thunderbird

[madduck]

Invoked on a signed message with attachments, muttdown creates a MIME tree as follows:

I  1 <no description>           [multipa/alternativ, 7bit, 111K]
I  2 ├─><no description>        [multipa/signed, 7bit, 110K]
I  3 │ ├─><no description>      [multipa/mixed, 7bit, 109K]
I  4 │ │ ├─><no description>    [text/plain, quoted, utf-8, 0,4K]
A  5 │ │ └─>Brochure.pdf        [applica/pdf, base64, 108K]
I  6 │ └─>PGP signature         [applica/pgp-signat, 7bit, 1,1K]
I  7 └─><no description>        [text/html, quoted, utf-8, 0,7K]

This works fine with mutt, obviously, and Gmail also seems to be okay with it, but Thunderbird doesn't give access to the attachments, and in a way that makes sense, because I am advertising the text/html part that contains no attachments as a better alternative (better because later in the tree, cf. RFC 1341) to the whole multipart/signed container.

Short of duplicating the attachments or invalidating signatures, can you think of another means to address this?