Nonexistence of a Probability for Real Wave Equations
After a long search in the literature I must conclude that the issue raised by Bohm has attracted no interest, maybe because no one has ever entertained some doubt about the impossibility of building quantum mechanics with a real wave function.
I tried to give two precise mathematical formulations of Bohm's nonexistence statement in my post Conserved Current for a PDE, but, as I argued in my answer there, neither of the two seems to be a faithful mathematical translation of Bohm's physical statements.
Maybe, we will never know what physical and mathematical argument Bohm had in his mind not whether he had actually one. He could have maybe quoted without hesitation in his book some remark made by R.J. Oppenheimer (whose lectures at the University of California at Berkeley inspired a large part of Bohm's treatise) or he simply stated something he considered intuitively evident without worrying about a possible proof. We cannot know how the mind of a genius works ... and David Bohm was an absolute genius!
Looks like Bohm's proof deals with a free particle, which is not very realistic. However, if you consider the Schrödinger or Klein-Gordon equation, say, in electromagnetic field, you can make the wave function real by a gauge transform (at least locally), as noticed by Schrödinger in Nature (1952), v.169, p.538. Just one real function can also be sufficient for the case of the Dirac equation in electromagnetic field, a shown in my article in J. Math. Phys.