In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, non-well-founded set theory is also of practical interest, since many of the phenomenon that we wish to model are circular or non-well-founded in nature.
The anti-foundation axiom can be stated in a number of ways, including the following.
Every system of equations $\varepsilon$ has a unique solution.
Every graph $G$ has a unique decoration.
This axiom, in all the formulations of it that I have found, seems to me to be decidedly non-constructive, since the axiom simply asserts that there exists a solution or a decoration, but does not give any way to actually construct these. However, in practice, such a solution often can be found (for instance using Tarski's fixed point theorem), so my question is simply: does the wider mathematical and philosophical community consider the anti-foundation axiom to be constructive or not, or is it perhaps still a fairly open question?