Can you fast in the state of Junub if you woke up after Fajr?

Let's say you had marital relations and then you didn't have a ghusl and you woke up and Fajr has now started. Can you still fast? The answer is yes. Your fast is still valid. In fact, this is narrated at the time of the Prophet ﷺ in Sahih Bukhari. So as long as your marital relations finished before Fajr started, that's the key aspect, before Fajr started, then even though you hadn't taken a ghusl yet, you would still be able to fast. Of course, you take a ghusl, you pray Fajr and so on, and the fast is valid. The key aspect being no marital relations, no eating and drinking once Fajr has started.