Why does the Loanable Funds Market model use the real interest rate instead of the nominal interest rate? As far as I understand, the majority of loan contracts specify a nominal interest rate, NOT a real interest rate. So a hypothetical loanable funds markets would have people suggesting potential borrowers different loan contracts with different nominal intrerest rates.
As Brian mentions in his comments this is because in standard models when people make intertemporal decisions they take inflation into the account. Take the following example:
You have a rational utility maximizing person who makes decision between consuming $\\\$100$ today or saving it for later consumption. For simplicity lets assume discount factor is exactly 1 (this is unrealistic as clearly consumption in the future should be valued less than in present but it just simplifies the math). Now let us suppose the nominal interest rate $i$ is $10\%$ and the question is should you save in this example or not?
Well the correct answer here is we dont have enough information to say yes or no. If inflation is $5\%$ the person should save those $\\\$100$ if inflation rate is $15\%$ the person should not do that because in the former case you will increase your resources in real terms after accounting for inflation and in the latter you will have less of them. Hence nominal interest rate can be any arbitrary number $x$ between $[0,\infty)$, but  without knowing what inflation rate is we cannot say whether you should consume today or tomorrow (i.e. whether you should supply loanable funds to the market or not).
What matters is the real interest rate which is the nominal interest rate less the inflation rate since by Fisher equation: $r \approx i - \pi$ where $r$ is the real interest rate, $i$ nominal interest rate and $\pi$ inflation.
This covered just the supply side of loanable funds but the demand side would work in similar way. A business person would want to take loan for projects should analogously work with real returns and interest rates when evaluating business proposals.
I would not call this handwaving as Brian does which is commonly used in science to refer to addition to the model that are ad hoc or do not follow from assumptions of the model in consistent fashion but Brian is correct in saying that it is not necessarily exactly how reality works.
For example, it is documented that people can suffer from behavioral bias 'money illusion' which is cognitive bias to think of money in nominal, rather than real, terms. Under a 'money illusion' what the nominal rate is also matters in non-trivial ways. However, in simple macro 101 model which presents the classical view of  loanable funds market  where agents are rational forward looking and having rational expectations such money illusion does not exist.