Dialectal Arabic (DA) varieties are under-served by language technologies, particularly large language models (LLMs). This trend threatens to exacerbate existing social inequalities and limits language modeling applications, yet the research community lacks operationalized LLM performance measurements in DA. We present a method that comprehensively evaluates LLM fidelity, understanding, quality, and diglossia in modeling DA. We evaluate nine LLMs in eight DA varieties across these four dimensions and provide best practice recommendations. Our evaluation suggests that LLMs do not produce DA as well as they understand it, but does not suggest deterioration in quality when they do. Further analysis suggests that current post-training can degrade DA capabilities, that few-shot examples can overcome this and other LLM deficiencies, and that otherwise no measurable features of input text correlate well with LLM DA performance.