Acoustic interference of non-collinear waves

I’m getting wrong intensity for interference of sound waves arriving from different directions.

Suppose two identical coherent sound sources produce waves of intensity $I_0$ at a point $O$. Let the two waves arrive at $O$ in phase, but let their propagation directions be perpendicular. If I use the usual interference formula, $I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi$, then for $I_1=I_2=I_0$ and $\Delta \phi$, I get $I=4I_0$.

But I get a different answer if I start from the acoustic energy flux $\mathbf I=\langle p\mathbf u\rangle. $ Treating each wave as a locally plane progressive wave, the particle velocities are along their respective propagation directions. Since the directions are perpendicular, $u_1 \perp u_2$. At the same time, pressure is a scalar, so $ p=p_1+p_2. $ When I calculate $I$ directly, I seem to get $2\sqrt2 I_0$, rather than $4I_0$. So I'm not sure where the discrepancy is coming from.

If both waves can be treated as locally plane progressive waves, why don't the two methods give the same intensity?

Is the familiar $I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi$ formula implicitly assuming that the waves are collinear? More generally, what is the correct interference/energy-flux expression when two sound waves meet at an arbitrary angle?

I'd particularly appreciate a derivation starting from $\mathbf I=\langle p\mathbf u\rangle. $, because that's where I seem to be getting a different result.