This patient most likely has eosinophilic granulomatosis with polyangiitis (EGPA; formerly known as Churg-Strauss syndrome). EGPA is characterized by eosinophilia, migratory pulmonary infiltrates, purpuric skin rash, and mononeuritis multiplex in the setting of antecedent atopy. Although EGPA is considered an ANCA-associated vasculitis (specifically, antimyeloperoxidase/p-ANCA), 40% of patients with EGPA are negative for ANCA. This patient has involvement of the lungs (likely capillaritis with hemoptysis), nerves (mononeuritis multiplex), and skin. In the presence of systemic eosinophilia, elevated serum IgE levels, and a history of asthma, this patient's most likely diagnosis is EGPA. Other systemic multiorgan system diseases are less likely. For example, granulomatosis with polyangiitis (GPA; formerly known as Wegener granulomatosis) and microscopic polyangiitis would be unusual in the absence of ANCA and the presence of eosinophilia (and, in the case of GPA, lack of sinus involvement). Diagnosis of EGPA can be made most definitively by tissue biopsy. Sural nerve biopsy is most likely to yield pathology specific for EGPA, specifically, the presence of necrotizing vasculitis with eosinophilic granulomas. In addition, confirmation that the patient's neurologic findings are due to EGPA is important for characterizing the extent of disease and the level of immunosuppression needed. Such a biopsy would also allow the exclusion of other eosinophilic diseases, such as hypereosinophilic syndrome.

Although cryoglobulinemia commonly affects the nerves and skin (along with the kidneys, which are not involved in this patient), it uncommonly affects the lungs. Furthermore, the high level of eosinophilia is inconsistent with cryoglobulinemia.