Ephemeral key is held constant in ElGamal

Is there a known algorithm to recover the sender's messages in ElGamal $\mathbb{F}_p^*$ if the ephemeral key is held constant in two or more transmissions, assuming the messages are always distinct?

Say Alice picks a prime $p$ between $2^{1000}$ and $2^{1001}$ and chooses a $g \in \mathbb{F}_p^*$ with large prime order. $L$ is her public key. Doug the doofus sends Alice two or more transmissions $(c_1, c_i), c_1 = g^k (mod\,p), c_i = m_{i}L^k (mod\,p) i = 1,2,...,n$ with $k$ held constant. Can Eve the eavesdropper recover Doug's messages?

IOW, why does the ephemeral key need to be ephemeral?

• If $m_1 = m_2$ then the two ciphertexts will be identical, which Eve can certainly see. Even if not, Eve can compute $m_2/m_1$ or similar combinations. It doesn't directly leak $m_1$, but if Eve ever sees $m_2$ later on then she can recover $m_1$ from the quotient.
– user2552
Commented Jul 27, 2015 at 14:50

That is, she is given both $c_2 = mL^k$ and $c'_2 = m'L^k$; she can then compute the value $c_2\cdot c'^{-1}_2 = m\cdot m'^{-1}$.