Yes, it is possible. National Botanical Research Institute Phosphate (NBRIP) broth containing tri-calcium phosphate as the P source and can be use to explore high P solubilizing potential of any species, which can be exploited for the stabilization of fixed phosphate present in both soil and plant roots and also you can compare the phosphate solubilizing potential by different organisms.
With respect, the responses seem to mainly relate to the release of P from soil in the rhizosphere by excreted organic acids than to chemical reactions in a nutrient broth and their possible effect on uptake, which I think was the question
I think, the organic acids coming from bacterial cells will be present in broth solution as their conjugate base which will be negatively charged. Again the phosphate solubilized by them will also be of negatively charged. Then how is it possible, that two negatively charged moiety binding to each other, ionic bond is impossible. But if they behave like legand, then some kind of complex formation is possible in presence of metal ion.