Implications of $\mathsf{P}\neq\mathsf{NP}$ in $\mathsf{BSS}$ model
$\newcommand\Ptime{\mathsf P} \newcommand\NP{\mathsf{NP}} \newcommand\poly{\mathsf{poly}}$
It is known that $\Ptime/\poly \neq \NP/\poly \implies \Ptime_{\mathbb C}\neq \NP_\mathbb{C}$ [1] where the latter two represent the classes $\Ptime$ and $\NP$ in the BSS model over the complex numbers. (You can take the contrapositive to obtain a consequence of $\Ptime_\mathbb C = \NP_\mathbb C$.) 
To the best of my knowledge, this is the only known result, though results of the same kind are known for other classes (in particular in Koiran's weak model). 

For more on this, look at Chapter 1 (in particular Fig. 1.1) and Chapter 8 of Bürgisser's book [2].

[1] F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther. On real
Turing machines that toss coins. In Proc. 27th ACM STOC, Las Vegas,
pages 335–342, 1995.
[2] Peter Bürgisser. Completeness and reductions in algebraic complexity theory, 1998.