  From any two median spaces $X,Y$, we construct a new median space $X
\circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show
that this construction is compatible with wreath products in the following
sense: given two finitely generated groups $G,H$ and two (equivariant) coarse
embeddings into median spaces $X,Y$, there exist a(n equivariant) coarse
embedding $G\wr H \to X \circledast Y$. As an application, we prove that
$$\alpha_1(G \wr H) \geq \min(\alpha_1(G),\alpha_1(H))/2 \text{ for all
finitely generated groups $G,H$,}$$ where $\alpha_1(\cdot)$ denotes the
$\ell^1$-compression. As an other consequence, we recover several well-known
theorems related to the Hilbertian geometry of wreath products from a unified
point of view: the characterisation of wreath products satisfying Kazhdan's
property (T) or the Haagerup property, as well as their discrete versions (FW)
and (PW).
