Among the following categories of artistic style, to which was the Academy most closely tied: neoclassicism, romanticism, impressionism, or realism?
The following answer assumes that the "Academy" referenced in the question refers to what was originally (c. 17th Century) the Royal Academy of Sciences in Paris, now commonly referred to as "the Academy" and comprising one part of the Institut de France.
The Academy is best said to be associated with Neoclassicism. One of the earliest and most prominent artists affiliated with the Academy was Jacques-Louis David, who was educated there and whose neoclassic style focused heavily on Greek and Roman figures and structures. Additionally, Jean-Baptiste Vallin de la Mothe designed the Academy of Arts structure employing the Neoclassical style.
Theater of the period associated with the Academy also reflected the Neoclassic influences of the time, with productions heavily influenced by the enduring legacy of Cardinal Richelieu, the most powerful figure in France, and later Louis XIV. Artistic endeavors adhered strictly to the Neoclassical mode, although other components of society better reflected the other schools, especially Realism.