That looks like a pretty old design of reference. Doesn't mean it is bad, but all the modern 8 pin references have a much easier way to add trimming that does not need a separate supply or as much stability.
It is not just that you need to negative supply - it is also that after adding the negative supply, you may not actually need it. It is only there for the case that the reference voltage needs to be adjusted down. If your regulator is 9.998V, you could adjust it with just a positive supply only.
To answer the question, you do not need -15V. -10V is better. I bet they used -15V for the simple reason that the engineer writing the app notes up had a -15V supply already available. To get a very stable -10V, you use a standard inverting opamp circuit connected to the 10V reference out. Since the power in the two opamp resistors will be identical, they will track thermally pretty well.
A better option might be something like the Maxim MAX6350CSA+ (about $11). 1ppm temp stability.
But you are probably going to say it is only 5V.
Ok no big problem. Get two of them and two plastic cases. Power each one up from a 9V battery (They need 8V or more for the supply) and put the two in series to get 10V. Plus you can compare them against each other to check for drift.
Richard.