  Let $F|\mathbb{Q}_2$ be a finite extension. In this paper, we construct an
RZ-space $\mathcal{N}_E$ for split $\mathrm{GU}(1,1)$ over a ramified quadratic
extension $E|F$. For this, we first introduce the naive moduli problem
$\mathcal{N}_E^{\mathrm{naive}}$ and then define $\mathcal{N}_E \subseteq
\mathcal{N}_E^{\mathrm{naive}}$ as a canonical closed formal subscheme, using
the so-called straightening condition. We establish an isomorphism between
$\mathcal{N}_E$ and the Drinfeld moduli problem, proving the $2$-adic analogue
of a theorem of Kudla and Rapoport. The formulation of the straightening
condition uses the existence of certain polarizations on the points of the
moduli space $\mathcal{N}_E^{\mathrm{naive}}$. We show the existence of these
polarizations in a more general setting over any quadratic extension $E|F$,
where $F|\mathbb{Q}_p$ is a finite extension for any prime $p$.
