  Given a metric measure space $X$, we consider a scale of function spaces
$T^{p,q}_s(X)$, called the weighted tent space scale. This is an extension of
the tent space scale of Coifman, Meyer, and Stein. Under various geometric
assumptions on $X$ we identify some associated interpolation spaces, in
particular certain real interpolation spaces. These are identified with a new
scale of function spaces, which we call $Z$-spaces, that have recently appeared
in the work of Barton and Mayboroda on elliptic boundary value problems with
boundary data in Besov spaces. We also prove Hardy--Littlewood--Sobolev-type
embeddings between weighted tent spaces.
