I have created a work-around using manually-created axes:
backgroundData = [100 75 80 150];
foregroundData = [10 30 25 64];
% This works because ax1 is created AFTER ax2!
ax2 = axes('Position',[0.1300 0.1100 0.7750 0.8150]);
ax1 = axes('Position',[0.1300 0.1100 0.7750 0.8150]);
% Plotting the two bars on their axes. The order here doesn't matter.
bar(ax2,backgroundData,1.0,'FaceColor','b')
bar(ax1,foregroundData,.3,'FaceColor','r')
% Need to move the Y-axis after plotting the foreground data for some
% reason.
ax2.YAxisLocation = 'right';
ylabel(ax2,'Background Axis')
ylabel(ax1,'Foreground Axis')
% Now we have to set the background of the front axis to be transparent.
set(gca, 'Color', 'None')