We describe a controllable and unbiased strong-coupling diagrammatic Monte Carlo technique that is applicable to a wide range of fermionic systems and spin models. Unlike previous strong coupling methods that generally rely on the Grassmannian Hubbard-Stratonovich transformation, our construction is based on Wick's theorem and a recursive procedure to group contractions into effective connected vertices that are nonperturbative in all local physics and can be calculated exactly. The resulting expansion is described by simple diagrammatic rules that make it suitable for systematic treatment via stochastic sampling. Benchmarks against numerical linked cluster expansion display excellent agreement.