A lot of this will depend on your specific contract and the NDAs / non-compete agreements you sign with your "big corporate" employer. Do not rely on "something you have read on the Internet", get a lawyer, show them what you have signed or will sign, and discuss whether your planned startup will potentially be in conflict with these agreements. It is not sufficient to just discuss this with your manager, they cannot guarantee that nobody higher up will take issue with your startup (and if they are smart they will probably anyway give you a blanket "no", or at least a "no comment", since this is the only way to cover their own behind).
In general, I would be nervous about this construction. Most companies I am aware of would never be happy with their own employees directly competing with them, either now or in the future. The only case where I have seen similar constructions be viewed positive is if the startup somehow provides value-added rather than competition to the company - say, if the company provides some general tooling and the startup builds specific plugins for specific use cases. And even then the risk looms large that the company pivots and suddenly considers the startup a risk to their own baseline. This can particularly happen if the startup turns out to be successful - it's easy for a company to be happy with somebody else working a niche that does not seem sufficiently high-profit to them, but if it turns out that there actually is substantial profit in this niche the view may change quickly.
