- For myself, ideal means it is composed of ideal gas particles. This means that the ideal gas law applies. This means that there is no accounting for particles interacting which is a grave mistake for liquids and solids, and often for gases too.
- $dU = dH - d(pV)$ is always true. In an real system however, the variables involved may take on different quantitative values for a given state point, thus the final change in eternal energy $\Delta U$ may be different.
In general $\Delta U = \Delta H -V \Delta p - p \Delta V - \Delta p \Delta V$
If pressure is constant during the change then $\Delta p \Delta V = V \Delta p = 0$ which then leads to the term
$$\Delta U = \Delta H - p \Delta V $$
If volume is constant during the change then $\Delta p \Delta V = p \Delta V = 0$ which would then lead to
$$\Delta U = \Delta H - V \Delta p $$
If neither volume or pressure is constant the simplest form is
$$\Delta U = \Delta H -V \Delta p - p \Delta V - \Delta p \Delta V$$
$$\Delta U = \Delta H - (p_f V_f - p_i V_i) $$
see Andrew's answer for more details on the last step.
And I do correct you, non-ideal systems maintain equilibrium just as ideal systems do. If they didn't, we wouldn't be able to use thermodynamics on them since thermo only applies to equilibrium.