We use real-number codes to compress statistically dependent sources and establish a new framework for distributed lossy source coding in which we compress sources before, rather than after, quantization. This change in the order of binning and quantization blocks makes it possible to model the correlation between continuous-valued sources more realistically and compensate for the quantization error partially. We then focus on the asymmetric case, i.e., lossy source coding with side information at the decoder. The encoding and decoding procedures are described in detail for a class of real-number codes called discrete Fourier transform (DFT) codes, both for the syndrome- and parity-based approaches. We leverage subspace-based decoding to improve the decoding and by extending it we are able to perform distributed source coding in a rate-adaptive fashion to further improve the decoding performance when the statistical dependency between sources is unknown. We also extend the parity-based approach to the case where the transmission channel is noisy and thus we perform distributed joint source-channel coding in this context. The proposed system is well suited for low-delay communications, as the mean-squared reconstruction error (MSE) is shown to be reasonably low for very short block length.