I couldn't help imagining how OO "objects" might be implemented in Clay

Your dispatch code now:

    record Square {
    side : Double;
    }

    record Circle {
        radius : Double;
    }

    variant Shape = Square | Circle;

    procedure show;
    overload show(x:Square) { println("Square(", x.side, ")"); }
    overload show(x:Circle) { println("Circle(", x.radius, ")"); }

My "Objectivizing":

    interface Shape {
        abstract procedure show(self);
    }

    record Square : Shape  {
        side : Double;
        show(self) { println("Square(", self.side, ")"); }
    }

    record Circle: Shape {
        radius : Double;
        show(self) { println("Circle(", self.radius, ")"); }
    }
    variant Shape = Square | Circle;

Here, "abstract" declaration in the parent would make the children's "show" automatically become an overload. 'self' is variable which would be automatically specialized to the containing record. I think the resulting code isn't excessively verbose.