equation solving - Nsolve gives solutions with a small complex part which can't be eliminated by chop command

I do see that what I'm asking has been asked many times, and I have checked the previous questions, but they didn't work for the specific problem I'm asking: I have six equations with six unknowns:

Sqrt[2] x[2] y[1] + x[1] (x[2] - Sqrt[2] y[2]) == 1

Sqrt[2] x[3] y[2] + x[2] (x[3] - Sqrt[2] y[3]) == 1

x[1] (x[3] + Sqrt[2] y[3]) == 1 + Sqrt[2] x[3] y[1]

x[1]^2 + y[1]^2 == 1

x[2]^2 + y[2]^2 == 1


x[3]^2 + y[3]^2 == 1

and I have tried the following code to calculate the solutions:

sol = NSolve[{eq21, eq22, eq23, eq24, eq25, eq26}, 
WorkingPrecision -> MachinePrecision] // N // Chop

the problem is, all the solutions includes a complex part of the magnitude of around $10^{-8}$, which can't be chopped off by mathematica. One of the solutions looks like:

{x[1.] -> 1., x[2.] -> 1., x[3.] -> 1.,   y[1.] -> 2.84087*10^-7 - 5.67727*10^-8 I,   y[2.] -> 2.84087*10^-7 - 5.67727*10^-8 I,  y[3.] -> 2.84087*10^-7 - 5.67727*10^-8 I}}

I know that the above system has at least one solution of $[1,1,1,0,0,0]$, so this does not make sense.

For this problem, I could use the Solve command to get the exact solutions, but I will need to take care of more complex coefficients, which is shown to be difficult to be handled by Solve.

So I wonder how to get rid of the complex part, and get the precision of the result to be good? For example, I got some solution that gives me x[i]=1, but y[i] as a small complex number, while it should be 0.

Thanks a lot!

Update I

@Bob, thanks a lot for your code, but somehow, mathematica gives me some different result, especially when I use Nsolve, which returns an empty set. Any idea of what's going on? and could you please give a little bit more explanation about your code to help me understand it?

@ Daniel, even though I do not understand what it means, this method works perfect for this problem, even if I have some complex coefficients. Due to the complexity of the set of equations, I probably need to use Nsolve rather than Solve, and this method fixs the problem perfectly.

Answer

eqns = {
   Sqrt[2] x[2] y[1] + x[1] (x[2] - Sqrt[2] y[2]) == 1,

   Sqrt[2] x[3] y[2] + x[2] (x[3] - Sqrt[2] y[3]) == 1,
   x[1] (x[3] + Sqrt[2] y[3]) == 1 + Sqrt[2] x[3] y[1],
   x[1]^2 + y[1]^2 == 1,
   x[2]^2 + y[2]^2 == 1,
   x[3]^2 + y[3]^2 == 1};

vars = Cases[eqns, x[_] | y[_], Infinity] // Union

(*  {x[1], x[2], x[3], y[1], y[2], y[3]}  *)


solns = Solve[eqns, vars];

Verifying solutions

And @@@ (eqns /. solns)

(*  {True, True, True, True, True, True, True, True}  *)

vars /. solns // Column

EDIT: Using NSolve

(solns2 = ((vars /. NSolve[eqns, vars]) /.
      z_?NumericQ :> Round[Chop[z, 10^-6], 10^-6] //
     Union) /. z_Rational :> N[z]) // Column

EDIT 2: Or use high precision calculations

(soln3 = vars /. NSolve[eqns, vars, Reals, WorkingPrecision -> 30] // Union //
     Round[#, 10.^-6] &) // Column