# Are measurement results only orthogonal?

Are all measurement operators on a quantum mechanical system defined by a Hilbert space, such that all possible post-measurement states are orthogonal? For example measuring a qubit in some orthonormal basis $\{|0\rangle,|1\rangle\}$. The possible outcome states after measurement are $|0\rangle$ and $|1\rangle$. I know the example I gave above is projective measurement, a special case of general measurement. So is there an example where all possible post-measurement states are not orthogonal?

I know if the measurement operators are $\{M_m\}$ ( $m$ denotes a possible outcome) then if the outcome is $m$ post-measurement state is $\frac{M|\psi \rangle}{|M|\psi \rangle|}$ ( $|\psi\rangle$ being initial state of system ) such that $\sum M^{\dagger}_m M_m=I$. Thus I can mathematically see that the example I am looking for possible , but I can't come up with one having some physical significance.