It is not enough to state that something is a construction. When something is constructed, it is constructed from something else. There is always an underlying system of components (recursion is possible, but at some point, there must be something else) and there is always a process of perception or interpretation. That process implements the object in terms of the underlying objects and it is itself implemented in terms of something else (like neurons, or a programming language or a culture or society or institution, that in turn can be analyzed…).
It is OK to say “I am not interested in the implementation, just in the constuctedness”. However, if you do analyze the implementation of things, you will eventually “hit the hardware”. There must be something that pre-exists at some level, even if our access to it migt only be through some kind of language or notation and through a theory that is a construction. In that sense, consequent constructivism requires a realist basis.