Question: A complicated system of nonlinear equations

This system is from the last century. This one has already appeared several times on the forum.
It is clear that nowadays it is not difficult to find several of its solutions, especially with the help of Maple. But in this case we are talking about a complete solution in real numbers. Of course, there are techniques that practically allow us to talk about such a solution, only without theoretical justification. 
The simple idea of ​​transforming it from transcendental to polynomial and getting a complete solution on a theoretical basis did not lead to a good result: it turned out that this is too complicated work for the  Isolate procedure. Perhaps I made a mistake somewhere, or (and) my old PC is too weak.
(The values ​​of the constants are not very important.)

restart; with(RootFinding):
 b1 := 114.069^2; b2 := 109.2389^2; b3 := 103.892^2; b4 := 99.76348^2; b5 := 97.24296^2; 
f1 := x1^2+x2^2+x3^2+2*(x1*x2*cos(x4)+x2*x3*cos(x5)+x1*x3*cos(x4+x5))-b1; 
f2 := x1^2+x2^2+x3^2+2*(x1*x2*cos(2*x4)+x2*x3*cos(2*x5)+x1*x3*cos(2*(x4+x5)))-b2; 
f3 := x1^2+x2^2+x3^2+2*(x1*x2*cos(3*x4)+x2*x3*cos(3*x5)+x1*x3*cos(3*(x4+x5)))-b3; 
f4 := x1^2+x2^2+x3^2+2*(x1*x2*cos(4*x4)+x2*x3*cos(4*x5)+x1*x3*cos(4*(x4+x5)))-b4; 
f5 := x1^2+x2^2+x3^2+2*(x1*x2*cos(5*x4)+x2*x3*cos(5*x5)+x1*x3*cos(5*(x4+x5)))-b5; 
f := seq(cat(f, i), i = 1 .. 5):
# fsolve([seq(f[i], i = 1 .. 5)]);
for i to 5 do 
F[i] := expand(f[i]);
F[i] := subs(cos(x4) = x4, cos(x5) = x5, F[i]); 
F[i] := subs(sin(x4) = sqrt(-x4^2+1), sin(x5) = sqrt(-x5^2+1), F[i]);
F[i] := subs(sqrt(-x4^2+1) = y4, sqrt(-x5^2+1) = y5, F[i]); 
F[i] := collect(F[i], [y4, y5]); 
F[i] := subs(y4 = sqrt(-x4^2+1), y5 = sqrt(-x5^2+1), F[i]); 
F[i] := op(1, F[i])^2-(sum(op(k, F[i]), k = 2 .. nops(F[i])))^2 
end do:
for i to 5 do 
F[i] 
end do;
#fsolve([seq(F[i], i = 1 .. 5)]);
#T := Isolate([seq(F[i], i = 1 .. 5)], [x1, x2, x3, x4, x5]): j := nops(T);
# T;

Edited: "+" to "-".