Antibodies are no more than big chemicals. And like all chemicals they can react with a range of other chemicals. Maybe a narrow range compared to the million chemicals known to exist but nonetheless antibodies are far from monogamous. Antibodies are also called gamma globulins and the more antibodies you have the more the chance there will be a reaction outside whatever caused the antibody in the first place. And despite being immune deficient AIDS patients have a plethora of antibodies. So positive HIV tests could be caused by antibodies which have a chemical but no biological or causative relationship with the proteins in the antibody tests. There is certainly no evidence to counter this possibility.
Why would AIDS patients generate such antibodies? Apart from those which are autoantibodies, they may arise because AIDS patients are infected with many infectious agents. For example, fungi and mycobacteria account for 88% of all AIDS diagnoses. There is plenty of evidence that antibodies directed against these agents can and do react with the proteins in the antibody tests.
This has been best shown by Kashala from Uganda in 1995. This discovery led to caution using the HIV ELISA and Western blot antibody tests in mycobacterial prevalent areas. Such as Africa where TB is rife. In fact in their paper Kashala and his coworkers published a series of Western blots from non-HIV-infected leprosy patients (another mycobacterium) which would be reported HIV positive and infected in the US or Europe. Or even in Australia which has the most stringent diagnostic criteria of all.
Further evidence that HIV test have nothing to do with a retrovirus come from a study done in the US in 1990.