Which is correct: 'The girl who you described as smart has failed the exam' or 'The girl whom you described as smart had failed the exam'?
There are two possibilities. In formal English we would say, or, especially, write: 'The girl whom you described as smart has failed the exam.' Whom always denotes the object of a verb and would be replaced by 'her' if we were to split this sentence into two separate sentences: 'The girl has failed the exam. And yet you described her as smart.' (Note that 'smart' = 'clever' in this context.)
In conversational English, 'in practice' as you say, the relative pronoun would normally be left out completely: 'The girl you described as smart has failed the exam.' (You will also in speech find people saying: 'The girl who you described as smart has failed the exam', although this is not grammatically accurate or formally correct.)
Omission of the relative pronoun whom is particularly common when the verb is linked to a preposition. Compare the following 'defining' relative clauses:
* 'The friend who I went out with last night is anorexic.' (Grammatically incorrect, but sometimes/often heard.)
* 'The friend I went out with last night is going to have a baby.' (Omission of the relative pronoun would be the most common occurrence in these sentences.)
* 'The friend with whom I went out last night has bought a new car.' (Quite improbable, almost impossible, because, as conversational English, it is far too clumsy.)
However, in this statement, which is much more formal, it is possible to link the preposition with whom:
* 'The senator, with whom I dined last night, will be the next President of the United States of America.'
Of course, if who is the subject, rather than the object, of the relative clause, or if the relative clause is 'non-defining', i.e. used to convey non-essential information, who cannot be omitted. We cannot say:
* 'That's the man used to live next door to us.'
* 'The woman in the fur coat used to live next door to us is now President of Bigfoot Engineering.'
We have to say:
* 'That's the man who used to live next door to us. ('That's the man. He used to live next door to us.')
* 'The woman in the fur coat, who used to live next door to us, is now President of Bigfoot Engineering.'
Note the essential punctuation which surrounds the non-defining relative clause, which, if it became two sentences, would read: 'The woman in the fur coat is now President. She used to live next door to us.'
There is one other possible relative pronoun which we could use with the 'friend I went out with' sentences and that is that which is often an alternative for 'whom', 'who' and 'which'. That is very versatile because it can refer to people or things and can be used as the subject or object of a relative clause. Compare the following:
* 'The friend that I went out with last night is going to get married.'
* 'Is this the sock that you're looking for?' (OR: 'Is this the sock you're looking for?')
* 'The girl that lives next door keeps parking her car on our drive.' (OR: 'The girl who...')
* 'These are the trees that blossom in February.' (OR: 'These are the trees which...')