Forcing ASP.NET WebAPI client to send a client certificate even when no CA match

Forcing ASP.NET WebAPI client to send a client certificate even when no CA
match

I have a specific application that requires the use of client certificates
for mutual authentication of HTTPS requests. The server has a flexible
certificate validation policy that allows it to accept self-signed client
certificates that are not present in the server's certificate store. This
is known to work just fine using curl as a client.
What I've determined via testing and packet sniffing is that Microsoft's
ASP.NET HttpClient tries to be overly smart during the SSL handshake. This
particular client will only use a client certificate (from the
WebRequestHandler.ClientCertificates collection) if it has chain of trust
to one of the server's trusted roots. What I've observed is that if there
are no certificates with chain of trust then the client simply won't send
a certificate at all during the handshake.
This is understandable default behavior but is overly restrictive and
there appears to be no way to turn it off. I have experimented with
various other WebRequestHandler properties, including AuthenticationLevel
and ClientCertificateOptions but to no avail.
Is there a way to force HttpClient to send a client certificate when it is
available in the ClientCertificates collection, even though it appears
that it will not validate on the server side? I'm open to both
straightforward and dirty (reflection hacks) solutions as I really need
this client to work.