  This paper develops algebraic geometry over Henselian real valued (i.e. of
rank 1) fields $K$, being a sequel to our paper about that over Henselian
discretely valued fields. Several results are given including: a certain
concept of fiber shrinking (a relaxed version of curve selection) for definable
sets, the canonical projection $K^{n} \times K\mathbb{P}^{m} \to K^{n}$ and
blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a
descent property for blow-ups, a version of the Lojasiewicz inequality for
continuous rational functions and the theorem on extending continuous
hereditarily rational functions, established for the real and $p$-adic
varieties in our joint paper with J. Kollar. The descent property enables
application of desingularization and transformation to a normal crossing by
blowing up in much the same way as over the locally compact ground field. Our
approach applies quantifier elimination due to Pas.
