quantum mechanics - Set of states ${|phi_nrangle}$ in the density operator $rho=sumlimits_n p_n|phi_nranglelanglephi_n|$

The set of quantum states $\{|\phi_n\rangle\}$ in the definition of the density operator $$\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$$ need not be orthonormal, and need not form a basis. But unfortunately, in the examples that I have seen so far, the states $\{|\phi_n\rangle\}$ were both orthonormal and forms a basis.

Example 1 In the Stern-Gerlach (SG) set-up, the state of the silver atoms coming out of the oven and before passing through the magnetic field, is imperfectly known because $S_z$ remained unmeasured. Therefore, on the ignorance ground, such an ensemble will be represented by $$\rho=\frac{1}{2}(|{\uparrow}\rangle\langle{\uparrow}|+|{\downarrow}\rangle \langle{\downarrow}|).\tag{1}$$ Note that, in this case, the states $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ are orthonormal and forms the $S_z$-basis.

Example 2 Consider an unpolarized light moving in the z-direction so that its polarization must be in the $xy$-plane. Since we do not know the state vector, it is described by the density operator $$\rho=\frac{1}{2}(|x\rangle\langle x|+|y\rangle\langle y|)\tag{2}$$ where $|x\rangle$ and $|y\rangle$ describe plane polarized states along the $x$ and $y$-axes respectively.

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Question Can someone suggest an example of a mixed ensemble where the states $\{|\phi_n\rangle\}$ need not be orthonormal and need not form a basis? I'm not looking for the trivial example where the desity operator describes a pure state.