I think we can formalize constructible structures as any mathematical structure which can be uniquely defined by at least one finite sequence of symbols.

As far as not wanting to pedantically refer to "constructible reals", I agree that doesn't sound fun.

The better solution would be having a clear common pithy term for "unconstructible reals", for:

1. Teaching math related to numbers, until unconstructible reals or other structures had any relevance. I.e. most people never, ever.

2. For talking about algorithms, physics and other constructible structures, where the term reals is pervasivably used to mean constructible reals.

3. Most students get introduced to the fact that the cardinality of reals is greater than the cardinality of integers. But would be surprised, and get more use, out of knowing that the cardinality of instantiatable numbers (the ones they could define, calculate, measure or apply in virtually every situation but highly abstract math games), is EXACTLY the same as the integers.

Un-constructible structures are an interesting but exotic concept that shouldn't be riding around sereptitiously in common vocabulary.

A fair number of responses here involved confusion about what "reals" covers. And this is HN.