This paper considers the problem of distributed lossy compression where the goal is to recover one or more linear combinations of the sources at the decoder, subject to distortion constraints. For certain configurations, it is known that codes with algebraic structure can outperform i.i.d. codebooks. For the special case of finite-alphabet sources, recent work has demonstrated how to incorporate joint typicality decoding alongside linear encoding and binning. This work takes a discretization approach to extend this rate region to include both integer- and real-valued sources. As a case study, the rate region is evaluated for the Gaussian case. The resulting joint-typicality-based rate region recovers and generalizes the best-known rate region for this scenario, based on lattice encoding and sequential decoding.