In collaborative creation tasks, people steer artifacts towards specific goals by _refining_ them with _multimodal_ communication over multiple rounds of interaction. In contrast, generative AI excels at creating artifacts in a single turn but can struggle to make precise refinements that match our design intent. To close this gap, we present mrCAD, a dataset of multi-turn interactions in which pairs of humans iteratively created and refined computer-aided designs (CADs). In each game, a _Designer sent instructions to a _Maker_, explaining how to create and subsequently refine a CAD to match a target design that only the _Designer_ could see. mrCAD consists of 6,082 communication games, 15,163 instruction-execution rounds, played between 1,092 pairs of human players. Crucially, _Designers_ had access to two communication modalities – text and drawing. Analysis finds that players relied more on text in refinement than in initial generation instructions, and used different linguistic elements for refinement than for generation. We also find that state-of-the-art VLMs are better at following generation instructions than refinement instructions. These results lay the foundation for modeling multi-turn, multimodal communication not captured in prior datasets.