Given a family of real probability measures $(μ_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(ν^α)_{α\in [0,1]}$ with $ν^α(\mathbb{R})=α$ that are submeasures of $μ_0$, called a parametrization of $μ_0$. Then, for any $α$ we define an evolution $(η^α_t)_{t\geq 0}$ of the measure $ν^α=η^α_0$ across our peacock by setting $η^α_t$ equal to the obstructed shadow of $ν^α$ in $(μ_s)_{s \in [0,t]}$. We identify conditions on the parametrization $(ν^α)_{α\in [0,1]}$ such that this construction leads to a unique martingale measure $π$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(ν_{\text{lc}}^α)_{α\in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.