Method SynthTextPaneUI.installUI(JComponent c) says:

    Installs the UI for a component. This does the following things.
    1. Sets the associated component to opaque ...

Actually it doesn't happen.
See the following code sample:
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import javax.swing.*;
import javax.swing.plaf.synth.*;

public class InstallUIComponentOpaque {

    public static void main(String[] args) {
        SwingUtilities.invokeLater(new Runnable() {
            @Override public void run() {
                doTheTest();
            }
        });
    }

    private static void doTheTest() {
        try {
            UIManager.setLookAndFeel(new SynthLookAndFeel());
        } catch (UnsupportedLookAndFeelException e) {
            throw new RuntimeException(e);
        }
        JTextPane textPane = new JTextPane();
        SynthTextPaneUI ui = new SynthTextPaneUI() {
            @Override
            public void installUI(JComponent c) {
                System.out.println("c.isOpaque() = " + c.isOpaque());
                super.installUI(c);
                System.out.println("InstallUIComponentOpaque.installUI called");
                System.out.println("c.isOpaque() = " + c.isOpaque());
            }
        };
        textPane.setOpaque(false);
        textPane.setUI(ui);
    }
}
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The output will be:

c.isOpaque() = false
InstallUIComponentOpaque.installUI called
c.isOpaque() = false