  We study quotients of principally polarized abelian varieties with real
multiplication by Galois-stable finite subgroups and describe when these
quotients are principally polarizable. We use this characterization to provide
an algorithm to compute explicit cyclic isogenies from kernel for abelian
varieties with real multiplication over finite fields. Our algorithm is
polynomial in the size of the finite field as well as the degree of the isogeny
and is based on Mumford's theory of theta functions and theta embeddings.
Recently, the algorithm has been successfully applied to obtain new results on
the discrete logarithm problem in genus 2 as well as to study the discrete
logarithm problem in genus 3.
