For typical first-order logical theories, satisfying assignments have a straightforward finite representation that can directly serve as a certificate that a given assignment satisfies the given formula. For non-linear real arithmetic with transcendental functions, however, no general finite representation of satisfying assignments is available. Hence, in this paper, we introduce a different form of satisfiability certificate for this theory, formulate the satisfiability verification problem as the problem of searching for such a certificate, and show how to perform this search in a systematic fashion. This does not only ease the independent verification of results, but also allows the systematic design of new, efficient search techniques. Computational experiments document that the resulting method is able to prove satisfiability of a substantially higher number of benchmark problems than existing methods.