The nominalist's identity theory holds that a predication such as 'Socrates is a man' is true just in case Socrates is identical with one of the (personal) supposita of 'man', whereas inherence theory holds that it is true just in case Socrates's human nature is actual. This suggests that the inherence theory brings with it an additional ontological commitment to a possible kind of spurious entity, namely, natures of particulars. Klima objects here and claims that inherence theory has no such commitment. Rather, it is committed only to the idea that natures be conceived in abstraction from singulars, and not that they exist in abstraction from singulars.
It is not entirely clear from what Klima says but one important difference, it seems, is that an inherent universal exist insofar as it inheres in particulars, and hence it cannot exist independently of those particulars. Klima says that universals for the medievals are "objects of the abstract intellect" [lvii], whereas it is clear for Plato that the forms are entirely independent of the intellect and even the things that instantiate them.
It would still be a significant ontological commitment for the realists to be committed to inherent universals. For one, it would violate Ockham's razor of "multiplying beings according to the multiplicity of terms". Klima proposes a way out. He says that the realist has this commitment only if the ultimate significata of terms like 'man' are taken to be distinct from their supposita and from each other. But we might say instead that when a ball of wax ceases to be a ball (because it's molded into a cube, e.g.), its shape, i.e. roundness, ceases to exist, we can say that its shape becomes cubicity because the shape of the wax "can be understood as being but a certain characteristic arrangement of the dimensions of the wax".
It seems to me that Klima is confusing "the shape of the wax", which contingently and just so happens to be round (at a particular time), with roundness. The shape of the wax contingently has no corners, but it is essential of roundness that it has no corners. So roundness cannot become cubicity even if the shape of the wax an become cubicity. In the jargon of Kripke, the error lies in the non-rigid designation of 'the shape of wax' to pick out roundness.
Even so, the realist would still be committed to at least one inherent universal that any existing thing has and that is subject to change over time. But as we've just noticed, since particular shapes have their intrinsic properties essentially, Klima's argument fails, and the realist is committed to the existence of a multiplicity of inherent universals.