Probabilistic programming languages (PPLs) provide language support for expressing flexible probabilistic models and solving Bayesian inference problems. PPLs with programmable inference make it possible for users to obtain improved results by customizing inference engines using guide programs that are tailored to a corresponding model program. However, errors in guide programs can compromise the statistical soundness of the inference. This article introduces a novel coroutine-based framework for verifying the correctness of user-written guide programs for a broad class of Markov chain Monte Carlo (MCMC) inference algorithms. Our approach rests on a novel type system for describing communication protocols between a model program and a sequence of guides that each update only a subset of random variables. We prove that, by translating guide types to context-free processes with finite norms, it is possible to check structural type equality between models and guides in polynomial time. This connection gives rise to an efficient type-inference algorithm for probabilistic programs with flexible constructs such as general recursion and branching. We also contribute a coverage-checking algorithm that verifies the support of sequentially composed guide programs agrees with that of the model program, which is a key soundness condition for MCMC inference with multiple guides. Evaluations on diverse benchmarks show that our type-inference and coverage-checking algorithms efficiently infer types and detect sound and unsound guides for programs that existing static analyses cannot handle.