The phrase "realise whether" is correct and usable in written English.
It can be used when discussing the process of understanding or determining a particular condition or situation.
Example: "I need to realise whether this project is feasible before we proceed."
Alternatives: "determine if" or "ascertain whether."
Amazon is the latest company to realise that, whether you are a foreign or domestic company, Scotland is a good place to invest.
At the very least, it should make intrepid travellers realise that, whether or not they forget their swimming costume or even their skis, the one thing they can't afford to leave without is enough money, whatever the currency.
Prime Minister David Cameron told MPs: "This is about psychopathic terrorists that are trying to kill us and we do have to realise that, whether we like it or not, they have already declared war on us".
Realise that whether or not you pass a class, does not define your self worth, and that a thousand F's don't make you any less awesome!
Whether they realise it or not, whether or not they even care, from now on thousands of cyclists are doomed to commute on a giant Barclays ad.
He realised that whether political regimes stand or fall, the minutiae of particular human lives matter most.
"As a child you realise quite early whether you're good at something.
That is so important in trying to create an environment where people are no longer bullied because of their sexuality - and where they can realise their potential, whether as a great mathematician like Alan Turing, a star of stage and screen like Sir Ian McKellen or a wonderful journalist and presenter like Clare Balding".
Whether they would realise that it's for real or whether they would think it was a joke.
But the space affects them too, whether they realise it or not.
