I haven't heard much about tariffs in relation to political parties for some time. It seems to me that regardless of one's position on the political spectrum, tariffs enter in the discussion only when politically expedient.
In general, industrialists prefer them as they are protection against cheap imports flooding the market while agriculturalists tend to be against them so it is easier to sell their stuff overseas because countries will fight tariffs with tariffs.
Reagan refused to put tariffs on auto imports during the 80s calling them "protectionist" and telling US auto makers, "If you want to compete with foreign cars, you need to build better cars affordably."
So, I suppose real Republicans might stray away from tariffs because they represent undue government influence on the market; a historic GOP position.