  Let $V$ be an algebraic variety defined over $\mathbb R$, and $V_{top}$ the
space of its complex points. We compare the algebraic Witt group $W(V)$ of
symmetric bilinear forms on vector bundles over $V$, with the topological Witt
group $WR(V_{top})$ of symmetric forms on Real vector bundles over $V_{top}$ in
the sense of Atiyah, especially when $V$ is 2-dimensional. To do so, we develop
topological tools to calculate $WR(V_{top})$, and to measure the difference
between $W(V)$ and $WR(V_{top})$.
