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Another alternative is to use JavaScript or PureScript (Haskell-like language that compiles to JS) to build a separate frontend that communicates JSON or HTML (https://htmx.org) to/from your backend. Miso was already recommended to you which is a great project, but working with GHCJS is a nightmare from my personal experience.
ghcjs is the way to go for you, and soon it might be asterius. i do not know how hard it is to set ghcjs up without a framework. but frameworks like obelisk (based on reflex-dom), shpadoinkle, and miso automate that for. i personally like obelisk for its functional reactive programming but it can get awkward and get in your way. so if gui programming is just a means to the end of this one small application and you are not really interested in it nor functional reactive programming, shpadoinkle or miso might suit you better. miso implements the elm architecture (also "TEA", "functional model view controller") and shpadoinkle implements something directly equivalent to the elm architecture. but shpadoinkle achieves more composable widgets by minimalizing the elm architecture. so i recommend shpadoinkle for its better concept although miso is more mature.
ghcjs is the way to go for you, and soon it might be asterius. i do not know how hard it is to set ghcjs up without a framework. but frameworks like obelisk (based on reflex-dom), shpadoinkle, and miso automate that for. i personally like obelisk for its functional reactive programming but it can get awkward and get in your way. so if gui programming is just a means to the end of this one small application and you are not really interested in it nor functional reactive programming, shpadoinkle or miso might suit you better. miso implements the elm architecture (also "TEA", "functional model view controller") and shpadoinkle implements something directly equivalent to the elm architecture. but shpadoinkle achieves more composable widgets by minimalizing the elm architecture. so i recommend shpadoinkle for its better concept although miso is more mature.
Not that I'm much of an expert, but if you're talking about a very barebones static single-page-app, then you could very easily get by just using blaze-html to put your elements on the page, and then scotty is a basic web framework you could use to serve up your app.
We have Haskell bindings for dear-imgui - https://hackage.haskell.org/package/dear-imgui. dear-imgui is basically the same thing as egui. I mention this because dear-imgui also has an extension that streams an application over websockets - https://github.com/ggerganov/imgui-ws. If we had Haskell bindings to this, then you'd be able to do exactly what is on the egui.rs website.