Tabes Dorsalis: A late manifestation of syphilis that affects the spinal cord, leading to symptoms such as intermittent numbness and weakness in extremities, erectile dysfunction, and other neurological deficits. The presence of syphilis in the question strongly points towards this diagnosis.

Diabetic Neuropathy: Diabetes is a common cause of neuropathy, which can present with numbness, weakness, and erectile dysfunction. HIV and syphilis can increase the risk of developing diabetes.
HIV-Associated Neuropathy: HIV can directly cause neuropathy, leading to symptoms like numbness, weakness, and pain in the extremities. Erectile dysfunction is also common in HIV patients due to the disease itself or antiretroviral therapy.
Vasculitis (e.g., due to syphilis or HIV): Both syphilis and HIV can cause vasculitis, which might lead to intermittent numbness and weakness by affecting blood flow to the nerves.

Multiple Sclerosis: Although less likely, multiple sclerosis (MS) can present with a wide range of neurological symptoms, including numbness, weakness, and sexual dysfunction. MS is not directly related to syphilis or HIV but is a critical diagnosis to consider due to its significant impact on quality of life and the availability of disease-modifying treatments.
Spinal Cord Compression: This is a medical emergency that can cause numbness, weakness, and sexual dysfunction. It could be due to various causes, including tumors or abscesses, especially in immunocompromised patients like those with HIV.

Sjögren's Syndrome: An autoimmune disorder that can cause neuropathy and might be associated with HIV, leading to symptoms like numbness and weakness. It's less common but should be considered in the differential diagnosis.
Amyloidosis: A condition characterized by the deposition of amyloid proteins in various tissues, including nerves, which can lead to neuropathy. It's rare but can be associated with chronic infections like HIV or syphilis.