plotting - Testing equality of graphics

Is there a way to reliably test the equality of two plots? Consider this:

plot1 = Plot[Sin[x], {x, 0, 2Pi}];
plot2 = Plot[Sin[x], {x, 0, 2Pi}];

You'd think these would be equal, but they're not.

Evaluate[plot1 == plot2]

SameQ doesn't work either:

Evaluate[plot1 === plot2]
(* False *)

At least this works:

Evaluate[plot1 == plot1]
(* True *)

Edit:

@eyorble provided a nice answer, but the plot thickens. How can we also get False when two plots are not equal?

plot3 = Plot[Cos[x], {x, 0, 2Pi}];
(plot1 == plot3) /. (x_String :> StringReplace[x, "Charting`Private`Tag$" ~~ __ -> "Charting`Private`Tag"])

Answer

plot1 == plot2 /. 
  (x_String :> StringReplace[x, "Charting`Private`Tag$" ~~ __ -> "Charting`Private`Tag"])

True

Plots seem to be otherwise deterministic, so that appears to be the only necessary change for comparison with ==.

To get False when they aren't equal, perhaps something like this would suffice:

Activate[Inactive[SameQ][plot1, plot2]
   /. (x_String :> StringReplace[x, "Charting`Private`Tag$" ~~ __ -> "Charting`Private`Tag"])]

Inactive and Activate are simply to ensure that SameQ doesn't evaluate before the replacement takes place.