What belief system would combine both elements from Calvinism and Arminianism? I've often heard some people describe themselves as a "Calminianist" in an attempt to say through a portmanteau that they ascribe to both Calvinism and Arminianism.

Is/Are there some universally acknowledged belief system(s) that combines the overlapping elements in Calvinism and Arminianism? Perhaps some system(s) that accept all points of both or 2 points of one and 2 points of the other, for example?
There are a few that attempt to do so. Both Calvinists and Arminians would say that any such attempt is internally inconsistent.

I don't know of any "universally acknowledged system" of theology that does so, except perhaps for Amyraldism, which denies limited atonement but affirms the other four points of Calvinism. (Any Protestant denying a point of Calvinism is necessarily affirming the corresponding point in Arminianism, and vice versa.) In non-confessional churches you'll find Calvinists and Arminians freely intermingling and you'll find one-, two-, three-, or four-point Calvinists. There is great diversity in Christendom on these things, but as I said, I don't think you'll find any system of theology seamlessly incorporating one element of Calvinism with one element of Arminianism.