Interesting question. In the thymus, CD4+CD8+ "double-positive" T cells are a normal phase of T cell development. In other tissues, their function (if they are real) is less clear. I don't know what type of experiments you are doing, however when analyzing tumor infiltrating lymphocytes in mouse models by flow cytometry, we have regularly found a small double positive population of T cells that increases or decreases in frequency depending on how we perform tumor tissue digestion - We don't know what to make of this but suspect it is at least partially artifactual. Please let me know if you discover anything interesting about peripheral (i.e. non-thymus) double-positive T cells.
I never dealt with T-cells but the double positive T-cells usually reside in the thymic cortex. These thymic cortical T-cells interact with MHC molecules for antigen presentation. So, I think it might depend on the tissue from where you are taking up. Because during T-cell development and selection where they are migrating you need to confirm first. And in your case which type experiment you are doing or which model you taking is really important.
We found that double positive T cells fraction (CD3+CD4+CD8+) in peripheral blood is a very common feature for lymphoma patients (HL and NHL). Some research groups have noticed those kind of T cells in biopsies from lymphoma's lesions and reactive lymph nodes.
A bit more information; these are in the intestinal mucosa of pigtailed macaques measured via flow cytometry and are CD3 positive. Additionally, their expression phenotypes are similar to CD4+ T cells in terms of memory and activation.