The choroid is a key vascular layer of the eye, supplying oxygen to the retinal photoreceptors. Non-invasive enhanced depth imaging optical coherence tomography (EDI-OCT) has recently improved access and visualisation of the choroid, making it an exciting frontier for discovering novel vascular biomarkers in ophthalmology and wider systemic health. However, current methods to measure the choroid often require use of multiple, independent semi-automatic and deep learning-based algorithms which are not made open-source. Previously, Choroidalyzer -- an open-source, fully automatic deep learning method trained on 5,600 OCT B-scans from 385 eyes -- was developed to fully segment and quantify the choroid in EDI-OCT images, thus addressing these issues. Using the same dataset, we propose a Robust, Resolution-agnostic and Efficient Attention-based network for CHoroid segmentation (REACH). REACHNet leverages multi-resolution training with domain-specific data augmentation to promote generalisation, and uses a lightweight architecture with resolution-agnostic self-attention which is not only faster than Choroidalyzer's previous network (4 images/s vs. 2.75 images/s on a standard laptop CPU), but has greater performance for segmenting the choroid region, vessels and fovea (Dice coefficient for region 0.9769 vs. 0.9749, vessels 0.8612 vs. 0.8192 and fovea 0.8243 vs. 0.3783) due to its improved hyperparameter configuration and model training pipeline. REACHNet can be used with Choroidalyzer as a drop-in replacement for the original model and will be made available upon publication.