DSolve Returns Incorrect Solutions for First-Order ODE

Bug introduced in 10.4 or earlier and persisting through 11.3. Reported to Wolfram Technical Support as CASE:4150361.

Fifty-one DSolve questions on this site are tagged with [bugs]. This may be the fifty-second. Consider (based on question 180227),

eq1 = y'[t] - Sqrt[(1 + y[t])/y[t]^2];
s = DSolve[{eq1 == 0, y == 1}, y, t]
(* {{y -> Function[{t}, (-4 + 4 I Sqrt + 4 (-12 Sqrt t + 9 t^2 +
Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(1/3) -
(-12 Sqrt t + 9 t^2 + Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(2/3) -
I Sqrt (-12 Sqrt t + 9 t^2 + Sqrt[-64 + 288 t^2 - 216 Sqrt
t^3 + 81 t^4])^(2/3))/(4 (-12 Sqrt t + 9 t^2 +
Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(1/3))]}, . . . *)

where the second solution is obtained by the first by t -> -t.

(y /. s[])[t] == ((y /. s[])[t] /. t -> -t)
(* True *)

Plotting the two solutions,

Plot[Evaluate@{ReIm[y[t] /. s[]], ReIm[y[t] /. s[]]}, {t, -3, 3},
PlotRange -> Full, PlotStyle -> {Blue, {Blue, Dashed}, Red, {Red, Dashed}},
ImageSize -> Large, AxesLabel -> {t, y}, LabelStyle -> {Bold, Black, Medium}] immediately suggests that the second solution is incorrect, because its slope is negative at the boundary condition, y == 1, whereas eq1 == 0 indicates that it should be positive. Verifying the two solutions by back-substitution, shows that the second solution is indeed incorrect near t == 0. Applying the boundary condition to an incorrect solution is meaningless, of course, and the entire second solution should be discarded. Moreover, the first solution is valid only to the left of the peak in the first curve in the first plot, i.e., for t <= 2 Sqrt / 3. DSolve should, therefore, have returned the single solution,

y -> ConditionalExpression[Function[{t}, (-4 + 4 I Sqrt + 4 (-12 Sqrt t +
9 t^2 + Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(1/3) -
(-12 Sqrt t + 9 t^2 + Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(2/3) -
I Sqrt (-12 Sqrt t + 9 t^2 + Sqrt[-64 + 288 t^2 - 216 Sqrt
t^3 + 81 t^4])^(2/3))/(4 (-12 Sqrt t + 9 t^2 +
Sqrt[-64 + 288 t^2 - 216 Sqrt t^3 + 81 t^4])^(1/3))],
t <= 2 Sqrt / 3]

So, is my assessment correct that this is a bug? And, is there any work-around apart from the manual effort just described?