"in a way that shows" is a correct and usable phrase in written English.
You can use it to describe how someone did something. For example, "He baked the cake in a way that shows his expertise."
Cut them in a way that shows off their shape.
Strings are printed in a way that shows the contents more clearly as characters.
But he is refreshing in a way that shows how [similar] the others are".
Mr. Reyes photographed the laborers "in a way that shows their dignity and humanity," seeking to depict their essence.
She's out-Twittered her enemies and connected with people in a way that shows the real Sarah.
And how can Western high art illustrate sex in a way that shows bodies with direct contact?
It answered an interesting question in a way that shows how courts can avoid creating opportunities for trial lawyers.
They have to respect that, but you have to do that in a way that shows them respect as well.
Set up the exhibit in a way that shows how the two cultures borrow artistic styles from each other.
We should be thinking about these "subjects" in a way that shows deeper interrelationships between concepts and "types" and that "interacts" and is "interoperable" with other systems.
While the exemplary voices may express paradox, self-contradiction, antithesis, even confusion, Shepperson arranges his examples in a way that shows readers an aggregate vision.
