Many interesting program properties involve the execution of \textit{multiple} programs, including observational equivalence, noninterference, co-termination, monotonicity, and idempotency. One strategy for verifying such \textit{relational properties} is to construct and reason about an intermediate program whose correctness implies that the individual programs exhibit those properties. A key challenge in building an intermediate program is finding a good \emph{alignment} of the original programs. An alignment puts subparts of the original programs into correspondence so that their similarities can be exploited in order to simplify verification. We propose an approach to intermediate program construction that uses e-graphs, equality saturation, and algebraic realignment rules to efficiently represent and build programs amenable to automated verification. A key ingredient of our solution is a novel data-driven extraction technique that uses execution traces of candidate intermediate programs to identify solutions that are semantically well-aligned. We have implemented a relational verification engine based on our proposed approach, called KestRel, and use it to evaluate our approach over a suite of benchmarks taken from the relational verification literature.