Models for FX exotics that respect the smile

Which models are used to price exotic FX options consistently with the observed volatility smile, and what are their advantages and drawbacks?

Local vol, stochastic vol, stochastic-local vol, jump models and vanna-volga. Local vol (Dupire) fits all vanillas exactly but gives unrealistic smile dynamics and flat forward smiles, mispricing barriers. Stochastic vol (e.g. Heston) has realistic dynamics but cannot fit the whole surface. Mixing the two (SLV) fits vanillas while controlling dynamics, at a computational cost. Vanna-volga is a fast market heuristic with weak theoretical grounding.