This paper is devoted to the analysis of the real interpolation of \(L^p\)-spaces of Banach space-valued vector measures on \(\delta\)-rings. In an interesting series of papers, the authors have studied the real and complex interpolation spaces of vector measures defined on \(\sigma\)-algebras; the finite measure case when scalar measures were considered. Actually, they showed the relevant role that is played by a new space – the Lorentz space of the semivariation of a vector measure – which seems to be crucial in the analysis of the interpolation spaces when vector measures are involved. In this paper, the authors extend the results to the case of vector measures on \(\delta\)-rings – the non-finite measure case –, showing in particular that, if \(\nu\) is such a vector measure being \(\sigma\)-finite locally strongly additive, \(0 < \eta <1 \leq q \leq \infty\) and \(1 < p_0 \neq p_1 < \infty\), then the real interpolation spaces satisfy \[ (L^{p_0}(\nu),L^{p_1}(\nu))_{\eta,q} = (L^{p_0}_w(\nu),L^{p_1}_w(\nu))_{\eta,q} =L^{p,q}(\|\nu\|), \] where \(1/p=(1-\eta)/p_0 + \eta/p_1,\) and \(L^{p,q}(\|\nu\|)\) is the corresponding Lorentz space. It is defined as the usual one, but the distribution function is defined using the semivariation \(\|\nu\|\), which is just a subadditive set function, instead of using a measure. Far from being a straightforward generalization of the interpolation of the spaces \(L^p\) for the scalar case, the results in this paper show that having a good representation of the real interpolation spaces is just a matter of having a good vector measure representation of the spaces involved and to know their convexity/concavity properties and their lattice properties (order continuity and Fatou property). This is a consequence of the well-known fact that under some Banach lattice type requirements, the class of order continuous \(p\)-convex Banach function spaces can be identified with the class of the spaces \(L^p(\nu)\) of adequate positive vector measures \(\nu\). I think the specialists in the field should take note of this fact.