Commonly claimed examples include Georgian and Esperanto.
One example of a phonemic orthography is the International Phonetic Alphabet, intended to accurately describe the pronunciation of a language. Other phonetic alphabets may be used for languages which have no standard written form; these orthographies are also phonemic.
Creating a phonemic orthography for a language such as English would be impossible, as pronunciations differ far too much. Given a standard dialect of a language it is not too difficult to create one, but it is inevitable that with time pronunciations will drift and either the orthography will have to be reformed or it will become non-phonemic - this is what happened to English and French, for example.
Note that phonemic orthographies are different from phonetic orthographies; whereas in a phonemic orthography, allophones will be represented by the same grapheme, a phonetic orthography demands, by its very nature, that the phonetically distinct allophones be written as such. To use an example from English, the "t" sound in the words "table" and "cat" would, in a phonemic orthography, be written with the same character; however, a phonetic orthography would make the distinction between the aspirated "t" in "table" and the unaspirated "t" in "cat."
In other words:
A phonemic orthography represents phonemes, the sounds distinguished by speakers of the language and used to differentiate words (e.g. "fat" vs "vat").
A phonetic orthography represents phones, the sounds humans are capable of producing, many of which will often be grouped together as a single phoneme in any given natural language, the groupings being varied across languages. English, for example, does not distinguish aspirated/unaspirated consonants, but other languages, like Hindi, do.