This paper studies capital reallocation in the fossil fuel industry by investigating who owns coal power plants -- the largest single source of global greenhouse gas emissions. We build a bottom-up measure of the ownership of these brown assets by merging asset-level data on firms‘ plant ownership (real capital) in Europe with firms‘ shareholder data (financial capital). We document a sharp increase in private firms‘ coal ownership since 2015, accompanied by a large decline in public equity ownership. A formal decomposition shows that the large decline in public equity ownership was however not due to capital reallocation ("exit") but to capital utilization: these investors scaled down plants, not sold them. Instead, state investors played a crucial role: they sold to private firms, while being the slowest at scaling down their plants. We illustrate the economics of brown capital allocation by calibrating a model in which asset owners vary in how they value externalities. The possibility of nationalization of coal plants by state investors that value social factors (e.g. jobs, "energy security") is an important limit to the ability of "green finance" to decrease aggregate emissions, in line with recent episodes in Germany and Poland.