A fast way to decide satisfiability of a set of simple fewnomial inequalities? Background
Considering a set of points $(x_i, y_i)$ in $\mathbb R^2$ and constraints between some triples of them, which state, whether the three points of the triple are oriented clockwise (R), counter-clockwise (L) or collinear (I), we can translate this into a set of inequalities of the form
$x_1 y_2 - x_1 y_3 - x_2 y_1 + x_2 y_3 + x_3 y_1 - x_3 y_2 < 0$
$x_1 y_2 - x_1 y_3 - x_2 y_1 + x_2 y_3 + x_3 y_1 - x_3 y_2 > 0$
$x_1 y_2 - x_1 y_3 - x_2 y_1 + x_2 y_3 + x_3 y_1 - x_3 y_2 = 0$
for (R), (L) and (I) respectively, where the polynomials on the left-hand side are obtained from the determinant
$\det \begin{pmatrix}
1 & x_1 & y_1 \newline
1 & x_2 & y_2 \newline
1 & x_3 & y_3
\end{pmatrix}
= \begin{pmatrix}
x_2 - x_1 \newline y_2 - y_1
\end{pmatrix}
\cdot \begin{pmatrix}
y_3 - y_1 \newline x_1 - x_3
\end{pmatrix}
$.

So the set of constraints is satisfiable iff the corresponding system of (in-)equalities is satisfiable.
Although this problem is NP-hard, i am interested in the following:

What is a (relatively) fast way to decide satisfiability of such a system of (in-)equalities?

Kind regards.
 A: I believe your problem is NP-hard.
If you had, not just triples-constraints for some triples, but had given the
orientation of every triple, then you have specified what is known as the combinatorial order type
of the point configuration. (See Handbook of Discrete and Computational Geometry, Chapter 5,
"Pseudoline arrangements.") This is equivalent (in some sense) to a rank-3 oriented matroid.
It is NP-hard to determine whether or not an oriented matroid is realizable by points in $\mathbb{R}^2$.  This was established in a 1991 paper by Peter Shor, "Stretchability of pseudolines is NP-hard"
(PDF link).
Jürgen Richter-Gebert's Ph.D. thesis was on this topic:
"On the realizability problem of combinatorial geometries—Decision methods," 1992.
Update. To respond to the request for more specific information, let me suggest
Günter Ziegler's 1996 article, "Oriented Matroids Today" (PDF link), Section 3.4, "Realization algorithms," where he says,

the most eﬃcient algorithm (in practice) currently available to ﬁnd a realization (if one exists) 
  is the iterative “rubber band” algorithm described in Pock [532].

As you can infer from the number 532, this article has a comprehensive bibliography!
I might also recommend looking at Aichholzer et al.'s 2001 paper,
"Enumerating Order Types for Small Point Sets with Applications,"
which includes a clear exposition of the relationship between order types and
pseudoline arrangements.  Oswin Aichholzer maintains a very nice Order Type web page, which
is worth visiting for the latest information.  There you will learn that there are exactly
2,334,512,907 order types of 11-point sets.
