According to the PPP, the RER should be constant over time but, in reality, it is not. See Figure 6 for some evidence on Germany, Japan and Mexico. PPP is based on the idea of goods arbitrage . The CPI of different countries are not comparable since they include very different goods. Even for homogenous goods, the PPP might hold very well.
Price discrimination is feasible only under some conditions.
Moreover, changes in the nominal exchange rates will affect the relative price of Japanese versus US cars. Exchange Rates and Inflation Figure 2.