Equality using Coercion considered harmful?

Currently, Sage specifies that, upon comparing with a==b two objects a and b that do not have the same parent, a coercion is attempted to put them in the same parent. To prepare a discussion on whether that specification is viable in the long run, we are gathering here (borderline?) use cases where this behaviour is harmful. (Note that there is also the related problem of broken notions of equality for elements with the same parent which is discussed at EqualityUnhashable.)

sage: bool(pi == 3.14159265358979323)
True
sage: hash(pi)
2943215493
sage: hash(3.14159265358979323)
1826823505

However Python 2.7 documentation specifies:

object.__hash__(self)
Called by built-in function hash() and for operations on members of hashed collections including set,
frozenset, and dict. __hash__() should return an integer. The only required property is that objects
which compare equal have the same hash value;

As a first consequence of the above behavior:

sage: {3.1415926535897932: 'approx', pi: 'exact'}
{3.1415926535897932: 'approx', pi: 'exact'}
sage: {0:"exact", 0.0000000000000000000:"approx"}
{0: 'approx'}

And also:

sage: pii = 3.14159265358979323
sage: bool(pii == pi)
True
sage: dd = {pi: "exact"}
sage: pi in dd
True
sage: pii in dd
False
sage: pii in dd.keys()
True

sage: R1 = RealField(2)
sage: R2 = RealField(56)
sage: pi1 = R1(pi); pi1
3.0
sage: pi2 = R2(pi); pi2
3.141592653589793
sage: pi1 == pi2
True
sage: bool(pi == pi2)
True
sage: pi1 == 3
True

sage: F1 = FiniteEnumeratedSet([0.000000])
sage: F2 = FiniteEnumeratedSet([0])
sage: F1 is F2
True
sage: F2.list()
[0.000000000000000]
sage:

sage: S = SymmetricFunctions(QQ)
sage: x = S.s()[5]
sage: y = S.p()(x)
sage: x == y
True
sage: hash(x), hash(y)
(-1840429907820881728, 5178019317311573726)

sage: bring=ZZ
sage: for a in cartesian_product([[bring.one()]]):
....:     print a[0].parent()
....:
Integer Ring
sage: bring=QQ
sage: for a in cartesian_product([[bring.one()]]):
....:     print a[0].parent()
....:
Integer Ring