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The assumption mechanism for symbolic expression is kind of broken, but you can select the real values as follows:

sage: [answer for answer in answers if all([value.rhs() in RDF for value in answer])]
[[x == -sqrt(6), y == 0, z == 0, u == 108],
 [x == sqrt(6), y == 0, z == 0, u == 108],
 [x == 0, y == -sqrt(6), z == 0, u == 108],
 [x == 0, y == sqrt(6), z == 0, u == 108],
 [x == -sqrt(3), y == -sqrt(3), z == 0, u == 27],
 [x == sqrt(3), y == sqrt(3), z == 0, u == 27],
 [x == -sqrt(3), y == sqrt(3), z == 0, u == 27],
 [x == sqrt(3), y == -sqrt(3), z == 0, u == 27],
 [x == 0, y == 0, z == -sqrt(6), u == 108],
 [x == 0, y == 0, z == sqrt(6), u == 108],
 [x == -sqrt(3), y == 0, z == -sqrt(3), u == 27],
 [x == sqrt(3), y == 0, z == sqrt(3), u == 27],
 [x == -sqrt(3), y == 0, z == sqrt(3), u == 27],
 [x == sqrt(3), y == 0, z == -sqrt(3), u == 27],
 [x == 0, y == -sqrt(3), z == sqrt(3), u == 27],
 [x == 0, y == sqrt(3), z == -sqrt(3), u == 27],
 [x == 0, y == -sqrt(3), z == -sqrt(3), u == 27],
 [x == 0, y == sqrt(3), z == sqrt(3), u == 27],
 [x == -sqrt(2), y == -sqrt(2), z == -sqrt(2), u == 12],
 [x == sqrt(2), y == sqrt(2), z == sqrt(2), u == 12],
 [x == -sqrt(2), y == -sqrt(2), z == sqrt(2), u == 12],
 [x == sqrt(2), y == sqrt(2), z == -sqrt(2), u == 12],
 [x == -sqrt(2), y == sqrt(2), z == -sqrt(2), u == 12],
 [x == sqrt(2), y == -sqrt(2), z == sqrt(2), u == 12],
 [x == -sqrt(2), y == sqrt(2), z == sqrt(2), u == 12],
 [x == sqrt(2), y == -sqrt(2), z == -sqrt(2), u == 12]]

Also, since you computation involves only polynomials, why not use them directly (here i work with real algebraic numbers) ?

sage: R.<x,y,z,u> = AA[]
sage: P1 = R(eq1.lhs())
sage: P2 = R(eq2.lhs())
sage: P3 = R(eq3.lhs())
sage: P4 = R(c1.lhs()-c1.rhs())
sage: I = R.ideal([P1,P2,P3,P4])
sage: I
Ideal (6*x^5 - 2*x*u, 6*y^5 - 2*y*u, 6*z^5 - 2*z*u, x^2 + y^2 + z^2 - 6) of Multivariate Polynomial Ring in x, y, z, u over Algebraic Real Field
sage: I.dimension()
sage: V = I.variety()
sage: V
[{x: -1.414213562373095?, z: -1.414213562373095?, u: 12, y: -1.414213562373095?},
 {x: 1.414213562373095?, z: -1.414213562373095?, u: 12, y: -1.414213562373095?},
 {x: -1.414213562373095?, z: -1.414213562373095?, u: 12, y: 1.414213562373095?},
 {x: 1.414213562373095?, z: -1.414213562373095?, u: 12, y: 1.414213562373095?},
 {x: -1.414213562373095?, z: 1.414213562373095?, u: 12, y: -1.414213562373095?},
 {x: 1.414213562373095?, z: 1.414213562373095?, u: 12, y: -1.414213562373095?},
 {x: -1.414213562373095?, z: 1.414213562373095?, u: 12, y: 1.414213562373095?},
 {x: 1.414213562373095?, z: 1.414213562373095?, u: 12, y: 1.414213562373095?},
 {x: 0, z: -1.732050807568878?, u: 27, y: -1.732050807568878?},
 {x: -1.732050807568878?, z: -1.732050807568878?, u: 27, y: 0},
 {x: 1.732050807568878?, z: -1.732050807568878?, u: 27, y: 0},
 {x: 0, z: -1.732050807568878?, u: 27, y: 1.732050807568878?},
 {x: -1.732050807568878?, z: 0, u: 27, y: -1.732050807568878?},
 {x: 1.732050807568878?, z: 0, u: 27, y: -1.732050807568878?},
 {x: -1.732050807568878?, z: 0, u: 27, y: 1.732050807568878?},
 {x: 1.732050807568878?, z: 0, u: 27, y: 1.732050807568878?},
 {x: 0, z: 1.732050807568878?, u: 27, y: -1.732050807568878?},
 {x: -1.732050807568878?, z: 1.732050807568878?, u: 27, y: 0},
 {x: 1.732050807568878?, z: 1.732050807568878?, u: 27, y: 0},
 {x: 0, z: 1.732050807568878?, u: 27, y: 1.732050807568878?},
 {x: 0, z: -2.449489742783178?, u: 108, y: 0},
 {x: 0, z: 0, u: 108, y: -2.449489742783178?},
 {x: -2.449489742783178?, z: 0, u: 108, y: 0},
 {x: 2.449489742783178?, z: 0, u: 108, y: 0},
 {x: 0, z: 0, u: 108, y: 2.449489742783178?},
 {x: 0, z: 2.449489742783178?, u: 108, y: 0}]