  Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper
is to compare its algebraic Witt group $W(V)$ with a new topological invariant
$WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the
sense of Atiyah) on the space of complex points of $V$, This invariant lies
between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear
topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$.
  We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and
$WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modulo
bounded 2-primary torsion. We give precise bounds for the exponent of the
kernel and cokernel of these maps, depending upon the dimension of $V.$ These
results improve theorems of Knebusch, Brumfiel and Mah\'e.
  Along the way, we prove a comparison theorem between algebraic and
topological Hermitian $K$-theory, and homotopy fixed point theorems for the
latter. We also give a new proof (and a generalization) of a theorem of
Brumfiel.
