  This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of $\sOB(N^{14})$ for the purely projection-based
method, and $\sOB(N^{12})$ for two subresultant-based methods: this notation
ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of
the polynomials. The previous record bound was $\sOB(N^{14})$.
  Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in $\sOB(N^{12})$, whereas the previous bound was
$\sOB(N^{14})$.
  All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
  Key words: real solving, polynomial systems, complexity, MAPLE software
