The ‘consortium agreement’ is an agreement between members of the consortium, to set out their internal arrangements for implementing the grant. It is purely internal. Partner organisations are not signatories to the grant agreement and have neither rights nor obligations under it. They do not form part of the consortium, and hence, the do not sign consortium agreement.
Partner organisations will, of course, have an interest in some/all project results (e.g. those developed by researchers while on secondment at their premises). Those issues will be however dealt with in a separate arrangement - partnership agreement. Partners do not have a general interest in all results, nor should they be involved in consortium decisions regarding exploitation or dissemination strategies: they are, after all, only third parties to the project.