The universe evolves according to the Schrodinger equation at all times. The non-unitary evolution of the wavefunction at a measurement is more a consequence of not taking qubits of the density matrix of the measuring device into account. Similar to the NOT gate being non-unitary but the CNOT gate being unitary due to ancilla qubits.

The question about why do we perceive that a universe in R rather a Hilbert space is because there is a non-trivial one to many mapping between the Hilbert space and many real worlds.