Trend and Value are pervasive anomalies, common to all financial markets. We address the problem of their co-existence and interaction within the frame-work of Heterogeneous Agent Based Models (HABM). More specifically, we extend the Chiarella (1992) model by adding noise traders and a non-linear demand of fundamentalists. We use Bayesian filtering techniques to calibrate the model on time series of prices across a variety of asset classes since 1800. The fundamental value is an output of the calibration, and does not require the use of an external pricing model. Our extended model reproduces many empirical observations, including the non-monotonic relation between past trends and future returns. The destabilising activity of trend-followers leads to a qualitative change of mispricing distribution, from unimodal to bimodal, meaning that some markets tend to be over- (or under-) valued for long periods of time.