Strategic shrouding of taxes by profit-maximizing firms can impair the effectiveness of corrective taxes. This paper explores tax shrouding and its consequences after the introduction of a digital sin tax designed to discourage harmful overconsumption of online sports betting in Germany. In response to the tax reform, most firms strategically shroud the tax, i.e., exclude tax surcharges from posted prices. Using an extensive web-scraped panel data set on online betting odds, I causally estimate the effect of the tax on consumer betting prices. Consumers bear, on average, 76%  of the tax burden. There is considerable and long-lasting heterogeneity in effects conditional on shrouding practices. Firms that shroud taxes can pass 90% of the tax onto consumers, while the pass-through rate is 16% for firms that directly post tax-inclusive prices. To understand the results' underlying mechanisms and policy implications, I propose an optimal corrective taxation model where oligopolistic firms compete on base prices and can shroud additive taxes. Tax shrouding is only attainable in equilibrium if (some) consumers underreact to shrouded attributes. According to the theoretical predictions, the empirically identified heterogeneity suggests that strategic tax shrouding significantly attenuates the positive corrective welfare effects of the tax. The results prompt regulating shrouding practices in the context of corrective taxation.