Incorrect result by DSolve

For real $x$ consider the trivial equation
$$|y'(x)|=-|x|.$$
Since the left side is always positive and the right always negative, there is no solution.
Let’s try

DSolve[Abs[y'[x]]==-Abs[x], y, x, Assumptions-> {x ∈ Reals}],

DSolve[Abs[y'[x]]==-RealAbs[x], y, x, Assumptions-> {x ∈ Reals}] 

and

DSolve[Sqrt[y'[x]^2]==-Abs[x], y, x, Assumptions-> {x ∈ Reals}] 

all giving the wrong result

{{y->Function[{x},Sign[x]/2 x^2+Subscript[[ConstantC], 1]]},{y->Function[{x},-Sign[x]/2 x^2+Subscript[[ConstantC], 1]]}} 

At least

DSolve[RealAbs[y'[x]]==-RealAbs[x], y, x, Assumptions-> {x ∈ Reals}] 

does return {}.

Is this a bug or a feature?

Note that this is just one example. In any case when the equation is $f(y'(x))=…$ and $f$ contains square root or absolute value the results are wrong.

Edit: Originally, the equation $|y'(x)|=-e^x$ was used for the example, but as a user found out, in that particluar case there is a complex solution.