The units on real exchange rates, therefore, are units of foreign good over units of domestic (home country) good, since real exchange rates show how many foreign goods you can get per unit of domestic good. (Technically, the home and foreign country distinction is irrelevant, and real exchange rates can be calculated between any two countries, as shown below.)
The example above illustrates this principle- if a bottle of US wine can be sold for $20, and the nominal exchange rate is 0.8 Euro per US dollar, then the bottle of US wine is worth 20 x 0.8 = 16 Euro. If a bottle of European wine costs 15 Euro, then 16/15 = 1.07 bottles of European wine can be purchased with the 16 Euro. Putting all of the pieces together, the bottle of US wine can be exchanged for 1.07 bottles of the European wine, and the real exchange rate is thus 1.07 bottles of European wine per bottle of US wine.
The reciprocal relationship holds for real exchange rates in the same way that it holds for nominal exchange rates. In this example, if the real exchange rate is 1.07 bottles of European wine per bottle of US wine, then the real exchange rate is also 1/1.07 = 0.93 bottles of US wine per bottle of European wine.