"Our tools are dialogical and existential debate"

I did at first think you might be able to use dialectic in place of dialogic, but I would actually suggest sticking to the more literal "dialogic" or "dialogical".
This is the difference according to Wikipedia(!):
In a dialectic process describing the interaction and resolution between multiple paradigms or ideologies, one putative solution establishes primacy over the others. The goal of a dialectic process is to merge point and counterpoint (thesis and antithesis) into a compromise or other state of agreement via conflict and tension (synthesis). "Synthesis that evolves from the opposition between thesis and antithesis."[1] Examples of dialectic process can be found in Plato's Republic.
In a dialogic process, various approaches coexist and are comparatively existential and relativistic in their interaction. Here, each ideology can hold more salience in particular circumstances. Changes can be made within these ideologies if a strategy does not have the desired effect. An example of the dialogic process can be found in Nozick's, Anarchy, State, and Utopia.
These two distinctions are observed in studies of personal identity, national identity, and group identity.

Although there is not much context really I think they are most likely using dialogism here.
HTH!

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Note added at 1 day7 hrs (2013-05-22 17:41:11 GMT) Post-grading
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Thank you Siobhan! Glad to have been of some help :-)