Castile and bastile are two terms that can mean anything and everything to many different people. I've seen soaps listed/made as castile or bastile but they contained very little olive oil.
Traditionally speaking true castile soap should only contain one oil and that's olive oil. The lather from a castile soap is one that people either love or hate. Castile soap is very gentle and highly moisturizing. The lather is very creamy rather than having big bubbles, but also many describe the lather as slimy.
The term bastile came along because people didn't like the lather of pure castile soap. So they added a few additional oils to boost the lather while keeping the high olive oil content of the soap. A good bastile soap will contain more than 60% olive oil but the more olive oil used the better.
Both terms castile and bastile can also be called a loose term because it can mean something different to different people. It might contain as little as 50% olive oil (or even less) or as much as 80% or more of olive oil. I've seen castile and bastile soaps recipes listed with as little as 25% olive oil.
But both castile and bastile soaps may contain liquids other than water. They may also contain other additives such as silk, clay, or even oatmeal. Fragrance and color may also be added to either soap to make them more desirable and unique to use.
Both of these soaps become much better as they age.
Since castile soap uses 100% olive oil I won't go into a recipe for that. But here are a few bastile soap recipe options. Remember to always run your recipe through a lye calculator such as soapcalc. Select the total weight of the oils you plan to use and the calculator will figure out how much of each ingredient you need to weigh out to complete your soap.
Olive Oil 75%
Coconut Oil 12%
Castor Oil 13%
Hardness 22 ~~ Cleansing 8 ~~ Conditioning 75
Bubbly Lather 20 ~~ Creamy Lather 26
Olive Oil 85%
Castor Oil 15%
Hardness 14 ~~ Cleansing 0 ~~ Conditioning 84
Bubbly Lather 14 ~~ Creamy Lather 28
Olive Oil 65%
Coconut Oil 20%
Castor Oil 15%
Hardness 27 ~~ Cleansing 13 ~~ Conditioning 70
Bubbly Lather 27 ~~ Creamy Lather 27
This gives you a few options to start with. Coconut Oil and Castor Oil are used because they are two of the most popular oils that add to the lather of the soap that is desired. Coconut oil adds a wonderful lather but if too much is used it can be very drying to the skin. Unlike castor oil which also makes a nice lather is very moisturizing.