Looking at the interface between B and A' and getting all information about B and A', you will not know that A' is metastable with respect to A. Proably it is not exactly meant like this.

Let's assume that you may achieve true equilibrium between A' and B, the (metastable) equilibrium transition temperature between B and A' is lower than of B->A. This is the reason that often rapid cooling is required to see A'. You have to bridge the temperature range where A may form but A' may not. This is easiest shown on a plot of the Gibbs energies vs temperature. Then the driving force (Gibbs energy difference) is typically smaller than for B->A' than for B->A. In order to observe B->A' typically the kinetics mast be faster for B->A' (Nucleation, faster diffusion, if necessary; more rapid interface velocity). Nucleation of A' may win, if e.g. the interface energy or the elastic energy terms are lower for A'.

I hopt this helps.

Thanks Andreas! So the interface doesn't sound very important for determining whether or not the stable or metastable phase will appear. Are there theories/models to predict which phase will appear if you know, for example, the thermal gradients / concentration gradients, interface velocity, etc in the material, along with the energy barriers associated with each phase? I've looked through some metallury books, and most discuss metastability as a function of cooling rate but it's all very qualitative or empirical.

When you refer to thermal gradients you propably refer to solidification processes. For these the temperature gradients are very important as the interface velocity is usually determined crucialy by the transport of the latent heat. Here, of course, special literature exist, but beyond what is taught in general lectures I cannot help much. Your reference to "metallurgy books" seems that there you looked at the soldification sections.

If you are more interested in solid-solid transformations, usually the thermal gradient is less important, simply becausee the latent heat associated with the transformations is much lower than for solidification (the invariant parameter is actually the transformation entropy, connected with the transformation enthalpy via the equilibrium transformation temperature). In that case there is a huge amount of concepts to describe phase transformation kinetics. Porter Easterling is a good starter (having also solidification sections). Models are of course on very different levels. If you want to find much more, you may refer to Christian.