Singing Accompaniment Generation (SAG) is the process of generating instrumental music for a given clean vocal input. However, existing SAG techniques use source-separated vocals as input and overfit to separation artifacts. This creates a critical train-test mismatch, leading to failure on clean, real-world vocal inputs. We introduce AnyAccomp, a framework that resolves this by decoupling accompaniment generation from source-dependent artifacts. AnyAccomp first employs a quantized melodic bottleneck, using a chromagram and a VQ-VAE to extract a discrete and timbre-invariant representation of the core melody. A subsequent flow-matching model then generates the accompaniment conditioned on these robust codes. Experiments show AnyAccomp achieves competitive performance on separated-vocal benchmarks while significantly outperforming baselines on generalization test sets of clean studio vocals and, notably, solo instrumental tracks. This demonstrates a qualitative leap in generalization, enabling robust accompaniment for instruments - a task where existing models completely fail - and paving the way for more versatile music co-creation tools. Demo audio and code: https://anyaccomp.github.io