If measuring them causes them to lose entanglement [...]
Measurement doesn't destroy entanglement. Quantum mechanics works through the unitary evolution of the wavefunction. Suppose that a pion decays into an electron and an antielectron with opposite spins. The spins are entangled. If the correlation between their spins were to somehow be erased, it would violate conservation of angular momentum. If Ed measures the electron's spin, and Alice measures the antielectron's, then, through a unitary process, Ed and Alice have become entangled as well. Ed and Alice are in a mixture of the states $\Psi_1=|\uparrow\downarrow\rangle$ and $\Psi_2=|\downarrow\uparrow\rangle$.
Now for practical reasons, we're not going to be able to observe interference between $\Psi_1$ and $\Psi_2$, because it's too hard to observe interference involving objects like human bodies. However, that doesn't mean that the state has become separable. Suppose that it did, and the result was that the system ended up in state 1. Then what would the time-evolution of the system look like in the basis consisting of the states 1 and 2? Since it takes any mixture of the electron-antielectron states 1 and 2 and sends it to a pure state 1, we could imagine that it would be represented by a matrix something like
1 & 0 \\
0 & 0
But this is a nonunitary matrix, which is impossible.