Question: Is there a proofreading solution that can integrate with my existing web application and provide real-time corrections?

WProofreader is a good option. It offers AI-based grammar and spelling checks in more than 20 languages. It can be integrated with web apps, HTML controls and WYSIWYG editors, and can be used with a browser extension or standalone API. Among its features are a multilingual sentence corrector with autocorrect and autocomplete suggestions, an AI paraphrasing tool, and strong security. It can be used in a variety of environments, including on-premise and cloud hosting, for high customization.

Another option is Ginger. It's an AI-based writing assistant that offers advanced grammar and spell checking, along with context-sensitive corrections and paraphrasing. It works on a variety of platforms, including Gmail, Facebook, LinkedIn and Microsoft Word, and offers a Ginger API for developers who want to build grammar and spell checking into their apps and websites.

LanguageTool also offers a range of language checking and paraphrasing tools. It works on more than 30 languages and dialects and offers plugins for a variety of writing tools and websites. Among its features are grammar checking, tone suggestions and paraphrasing options, making it good for professional and academic writers. Premium subscribers get extra features, like unlimited paraphrasing and advanced grammar checking.