Fusion-based hyperspectral image (HSI) super-resolution has become increasingly prevalent for its capability to integrate high-frequency spatial information from the paired high-resolution (HR) RGB reference (Ref-RGB) image. However, most of the existing methods either heavily rely on the accurate alignment between low-resolution (LR) HSIs and RGB images or can only deal with simulated unaligned RGB images generated by rigid geometric transformations, which weakens their effectiveness for real scenes. In this article, we explore the fusion-based HSI super-resolution with real Ref-RGB images that have both rigid and nonrigid misalignments. To properly address the limitations of existing methods for unaligned reference images, we propose an HSI fusion network (HSIFN) with heterogeneous feature extractions, multistage feature alignments, and attentive feature fusion. Specifically, our network first transforms the input HSI and RGB images into two sets of multiscale features with an HSI encoder and an RGB encoder, respectively. The features of Ref-RGB images are then processed by a multistage alignment module to explicitly align the features of Ref-RGB with the LR HSI. Finally, the aligned features of Ref-RGB are further adjusted by an adaptive attention module to focus more on discriminative regions before sending them to the fusion decoder to generate the reconstructed HR HSI. Additionally, we collect a real-world HSI fusion dataset, consisting of paired HSI and unaligned Ref-RGB, to support the evaluation of the proposed model for real scenes. Extensive experiments are conducted on both simulated and our real-world datasets, and it shows that our method obtains a clear improvement over existing single-image and fusion-based super-resolution methods on quantitative assessment as well as visual comparison. The code and dataset are publicly available at https://zeqiang-lai.github.io/HSI-RefSR/.