the phrase "indeterminate real number" could be seen as misleading since a number can't be indeterminate, while an indeterminate can't be a number. When we use $x$ in an equation like $ax+b=c$ we mean for it to serve as an indeterminate variable, one that can be replaced by any actual real number. The resulting graph is the graph of a straight line where we treat the parameters $a,b,c$ very differently than the variable $x$. However, in this context we do not mean that $x$ is an indeterminate real number. Rather we mean that it is a variable - a single indeterminate of real type - but not that it is a number.
On the other hand, if I say that choose a real number between $0$ and $1$ at random we usually envisage a situation where I actually pick such a number (so it is now chosen and can no longer change) but its value is unknown and thus is indeterminate, hence an indeterminate real number. But, it's not really indeterminate, it's just that we don't know what its value is. This is a slightly different situation that saying let $x$ be the real number between $0$ and $1$ that I'm going to choose in a minute. This time if we say that $x$ is an indeterminate real number then we can probably agree that it is indeed an indeterminate but few people will agree that it (currently) is a number at all.
Thus, the phrase indeed offers some challenges in interpretation as well as formalization.
The most common axiomatization of random variable is that a (real valued) random variable is a measurable function $x:M\to \mathbb R$ from some measure space $M$. This time calling such thing an indeterminate real number is objectionable since it's not a number at all, it's a function.
So, depending on the context, instead of saying "indeterminate real number" one can say "real variable" or "real random variable". Both are rigorous and offer very good axiomatizations for the concepts discussed above.