Decompilers are important tools for reverse engineers that help them analyze software at a higher level of abstraction than assembly code. Unfortunately, because compilation is lossy, deterministic decompilers produce code that is missing many of the details that make source code readable in the first place, like variable names and types. Neural decompilers, on the other hand, offer the ability to statistically fill in these details. Existing work in neural decompilation, however, suffers from substantial limitations that preclude its use on real code, such as the inability to define composite types, which is essential to fully specify function semantics. In this work, we introduce a new dataset, Realtype, that includes substantially more complicated and realistic types than existing neural decompilation benchmarks, and Idioms, a new neural decompilation approach to finetune any LLM into a neural decompiler capable of generating the appropriate user-defined type definitions alongside the decompiled code. We show that our approach yields state-of-the-art results in neural decompilation. On the most challenging existing benchmark, ExeBench, our model achieves 54.4% accuracy vs. 46.3% for LLM4Decompile and 37.5% for Nova; on Realtype, our model performs at least 95% better.