Client Side Authentication Between Two Servers?

Situation: All of our servers are required to use client side
certificates issued by a private authority.
Dilemma: Many of our web applications (and future, web applications
that we planned on developing) relied on connecting to another web
service *residing on another server* and pulling (or pushing)
information. At the moment, this can no longer be done because we
cannot connect to the other server without having valid authentication.

Is there anyway to get around this? Can you forward a client
certificate from one server to another? Can you assign a client
certificate to a server and have it automatically be used to access
other servers that require it?

Thanks!