In [Trans. Am. Math. Soc. 294, 243--262 (1986; Zbl 0602.20052)] \textit{N. R. Reilly} and the author studied the semigroups in the title, with respect to such properties as being E-unitary, fundamental, combinatorial and completely semisimple. This paper continues that study. Let \(\mathcal V\) be a variety of inverse semigroups: then \(F\mathcal V_X\) denotes the (relatively) free inverse semigroup in \(\mathcal V\), on the countably infinite set \(X\). A new notion introduced here is that of being ``completely fundamental'': \(F\mathcal V_X\) has this property if for any word \(a\) in the absolutely free inverse semigroup \(F\mathcal I_X\), and variable \(x\) not in the content of \(a\), comparability in \(F\mathcal V_X\) of (the images of) the idempotents \(zz^{-1}aa^{-1}\) and \(a^{-1}zz^{-1}a\) implies that (the image of) \(a\) is an idempotent in \(F\mathcal V_X\). This notion is stronger than fundamentality for \(F\mathcal V_X\), which is equivalent to the property obtained by replacing ``comparability'' with ``equality'' in the definition above. Moreover, if \(F\mathcal V_X\) is fundamental and also completely semisimple then it is completely fundamental. The example given in the cited reference, of a variety \(\mathcal V\) for which \(F\mathcal V_X\) is not completely semisimple, is fundamental but not completely fundamental. The paper contains many other results on E-unitariness, etc.