Elliptic curve trapdoor function without modular arithmetic?
First, as poncho correctly noted in a comment, real-based values don't work well with computers, because they require an infinite series / sequence representation and handling infinite amounts of data with a finite amount of storage usually doesn't end well, especially if you don't want to give away structure like in this case.

Second, picking keys is hard if you consider infinite ranges (even if there's an infinite amount of numbers between any two numbers), because you can't sample from an infinite set uniformly at random (then every element would have probability 0 of being picked).

Third, real-numbers often allow for more structure than we'd like. Take for example the logarithm. Computing $\log_3 b$ is easy in $\mathbb R$ (we know efficient algorithms that will yield a solid approximation) whereas finding this over $\mathbb Z_p$ tends to be hard. The fear is that a similar property can be found for the elliptic curve discrete logarithm problem. Also most problems over $\mathbb R$ are a matter of finding a good enough approximation where you can use intermediate approximations and have helpful navigatory metrics and thelike, whereas with crypto you want "hit-or-miss" and no guidance in the right direction.