I don't know what it is, but it sounds just like what my husband has and has had on his hands after working around moldy grains bins or rat poison bombs on a former job.
He went to several doctors before finally finding help at a ND. He gave him Nature's Sunshine Oregon Grape Root which is in a syrup form. He told him to rub some on his hands throughout the day and if it worked to take it internally. Well it started to work and he started taking it internally and it seemed to "solve" the problem, until he stopped taking it. It came back some but only after a long time after not taking the oregon grape. It also helped to clear up the chronic cough used to clear his throat.
I think he has a fungal infection from the moldy bins that needs some Diflucan, but who can get a doctor to prescribe Diflucan.