According to the LWP README.SSL , one may either use Crypt::SSLeay or IO::Socket::SSL to support SSL, though the former is recommended.

The Crypt::SSLeay documentation has this to say about how to use client certificates which may or may not be of use to you.

In reply to Re: HTTPS request with client authentication using certificates by bingos
in thread HTTPS request with client authentication using certificates by blackchicken

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