The source-coding problem with side information at the decoder is studied subject to a constraint that the encoder—to whom the side information is unavailable—be able to compute the decoder's reconstruction sequence to within some distortion. For discrete memoryless sources and finite single-letter distortion measures, an expression is given for the minimal description rate as a function of the joint law of the source and side information and of the allowed distortions at the encoder and at the decoder. The minimal description rate is also computed for a memoryless Gaussian source with squared-error distortion measures. A solution is also provided to a more general problem where there are more than two distortion constraints and each distortion measure may be a function of three arguments: the source symbol, the encoder's reconstruction symbol, and the decoder's reconstruction symbol.