A "radical" republican is a phrase coined to counter the phrase "radical" left. This done in an attempt to create a polar "opposite" group in the political "right", which is some how is supposed to be similar (but often time protrayed as worse), than those radicals or groups that blow stuff up, or cut peoples heads off, that often find shelter in the fringes of the "left".
The phrase radical "republican" is us largely used as a medium to create a paper tiger, or a straw man argument by the left, in compliance with the "Rules for Radicals" guidance. In essence the left accuses whoever the opposition is of the very thing that the left is in fact guilty of doing. This in effect is to "rob" those on the "right"of credibilty, and the "high" ground in the court of public opinion when pointing out the illegal or immoral activities of some on the left. This is especially effective when the audience is largely uninformed, and has no desire to be, which seems to be most of those on the left.