According to the equations of MHD, electromagnetic waves cannot propagate in a plasma if they are below the plasma frequency. (For more information, see this question about astronomical-wavelength radio waves and this question about the plasma frequency.) Thus, waves below the plasma frequency are reflected. That's why radio signals bounce off Earth's ionosphere.
Suppose we have a spacecraft in the interstellar medium and we start transmitting a directional beam of ELF radio waves at a frequency -- say, 1 kHz -- that's below the local plasma frequency. Since the beam carries momentum, doing this creates a small force on our spacecraft. Then, when the plasma reflects the beam back at our antenna, we can reflect the radio waves back again, pushing ourselves forward a little bit more. In principle, we could continue reusing the photons to provide even more thrust.
Would this effect let us create a photon rocket that beats the limits set by the (relativistic) rocket equation?
Another way of looking at this idea: naive photon rockets use the energy of a photon very "inefficiently." Suppose we build a nuclear photonic rocket that uses a fission reactor to power a laser beam. Let's say that the photons in the laser beam carry 1 eV each. In Earth's frame of reference, if the rocket is traveling at much less than the speed of light, the Doppler effect is small and those photons still appear to be carrying nearly 1 eV. But if we could recycle the photons, the Doppler effect would transfer some energy to the spacecraft each time the photons are reflected. Thus, the rocket could extract a much larger fraction of the energy in the laser beam.
For the sake of discussion, I'll try to be a bit more specific about what such a device might look like in practice.
The plasma density of the interstellar medium near the Solar System is thought to be about 0.1 cm$^{-3}$, which gives a plasma frequency of
$$ \frac{1}{2\pi} \sqrt{\frac{n_e e^2}{m_e \epsilon_0}} \approx \text{2.8 kHz} $$
(Voyager 1 measured a plasma frequency of 2.6 kHz, so this is in the right ballpark.) Pretending for a moment that the interstellar medium is a fully ionized plasma in thermal equilibrium at 7000 K (the approximate temperature of the Local Interstellar Cloud), we can use the Bohm-Gross dispersion relation to calculate the characteristic scale of a 1 kHz evanescent wave:
$$ \sqrt{\frac{3 k_B (\text{7000 K}) / m_e}{(\text{2.8 kHz})^2 - (\text{1 kHz})^2}} \approx \text{200 m} $$
(Applying some common sense to this, it's hard to believe that there are enough electrons in a region of this depth to actually reflect a high-power radio wave. I imagine that radiation pressure would clear out a substantial cavity in the plasma behind the spacecraft. However, the cavity walls should still be reflective.)
Let's say the mass of our spacecraft is $10^{6}$ kg and it has a 1 terawatt fusion reactor on board that converts deuterium and tritium into helium-4, extracting an average of 1 MeV from each reaction. If we discard the reaction products and use the reactor to power a laser beam, the spacecraft gains a momentum of $E/c = 0.0033 \frac{\text{eV} \cdot \text{s}}{\text{m}}$ per reaction while losing 5 amu of mass. For the purpose of the rocket equation, that gives an effective exhaust velocity of $1.3 \times 10^5$ m/s = $0.004c$ and a thrust of 3300 N. But if instead we use the reactor to power an ELF transmitter and, on average, we can get each photon to bounce back once from the interstellar medium and reflect off the spacecraft antenna, then we've just tripled both the thrust and the specific impulse. That brings us quite a bit closer to an interstellar-capable engine. So my question is, would this actually work from a physics perspective?
For the purposes of this question, I'm not interested in practical issues. Assume that we have the technical capability to build an absurdly lightweight superconducting dish antenna thousands of kilometers in diameter, a fusion reactor that can generate ridiculous amounts of power, and radiators that can reject arbitrary amounts of waste heat.
