Identify the wave number (k) from the given wave equation.

The general equation for a progressive wave is given by:

$$y = A\text{Sin}(\omega t - kx)$$

The given equation is:

$$y = 10\text{Sin}\left(1000\pi t - \frac{\pi x}{34}\right)$$

By comparing the two equations, we can see that the wave number $k$ is:

$$k = \frac{\pi}{34} \text{ rad/cm}$$

The phase difference $\Delta \phi$ between two points separated by a distance $\Delta x$ is given by the formula:

$$\Delta \phi = k \Delta x$$

We are given the separation distance $\Delta x = 153 \text{ cm}$.
Substitute the values of $k$ and $\Delta x$ into the formula:

$$\Delta \phi = \left(\frac{\pi}{34} \text{ rad/cm}\right) \times (153 \text{ cm})$$ $$\Delta \phi = \frac{153\pi}{34} \text{ radians}$$

We know that $\pi \text{ radians} = 180^\circ$.
So, to convert $\Delta \phi$ to degrees, we multiply by $\frac{180^\circ}{\pi \text{ radians}}$:

$$\Delta \phi = \frac{153\pi}{34} \times \frac{180^\circ}{\pi}$$ $$\Delta \phi = \frac{153 \times 180^\circ}{34}$$

Now, perform the multiplication and division:

$$\Delta \phi = \frac{27540^\circ}{34}$$ $$\Delta \phi = 810^\circ$$

A phase difference of $810^\circ$ means the wave has completed multiple cycles. To find the equivalent angle within a single cycle ($0^\circ$ to $360^\circ$), we subtract multiples of $360^\circ$:

$$810^\circ = 2 \times 360^\circ + 90^\circ$$

So, the phase difference is equivalent to $90^\circ$.

Comparing this result with the given options:
A. $270^\circ$
B. $45^\circ$
C. $90^\circ$
D. $180^\circ$
E. No Correct Option

The calculated equivalent phase difference matches option C.

The final answer is $\boxed{\text{C. } 90^0}$.