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You have the right idea: do f.change_ring(R) with the argument R set to one of K.embeddings(RR):

var('w')
K.<t> = NumberField(w^2-3)
R.<x,y> = PolynomialRing(K)
f = y - t*x
implicit_plot(f.change_ring(K.embeddings(RR)[0]),(x,-3,3),(y,-3,3))

You might also like:

implicit_plot(f.change_ring(K.embeddings(RR)[0]),(x,-3,3),(y,-3,3), color='blue') + implicit_plot(f.change_ring(K.embeddings(RR)[1]),(x,-3,3),(y,-3,3), color='red')