If I understand correctly then I should be able to use Stash to set a variable on Template A and then retrieve it on Template B.
Template A contains
{exp:stash:set name="testvar" scope="site"}foo{/exp:stash:set}
Template B contains
Var= {exp:stash:get name="testvar" scope="site"}
There is nothing else on these templates but I don't get the value for testvar output on Template B.