An Effective Criterion for Covering Maps Between Real Varieties

We prove that a quasi-finite flat morphism with locally constant geometric-fiber cardinality between varieties over a real closed field induces a covering map on the rational points. The algebraic core of the criterion is a characteristic-zero result showing that every such morphism becomes finite étale after reduction. Since reduction does not change rational points, the induced map is a covering in the Euclidean topology. We extend the covering conclusion to Boolean combinations of closed subschemes and their complements, and construct a finite stratification separating finite covering families from families with positive-dimensional fibers. Finally, we compute the canonical non-finite, non-finite-flat, and non-finite-étale loci, using Gröbner bases. This yields effective tests for the hypotheses of the covering criterion. We conclude with applications illustrating these constructions.