Why is egg white alkaline?
In the terms of the question, this can be regarded as calcium hydroxide, with the carbonate being in the form of calcium carbonate. Calcium hydroxide is a stronger base (pH of aqueous solution at 1mm = 11.27) than carbonic acid is an acid (pH of aqueous solution at 1mm = 4.68), so that 1mm calcium carbonate has a pH of 9.91. (Values from Aquion)
Calcium carbonate is, of course, the constituent of eggshell, which crystallizes out of solution when it is deposited in the uterus. Its presence within the egg white is presumably a consequence of this.
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I would add that one does not generally regard there being a particular base balancing the carbonic acid: when the hydrogen ions are removed there is an increase in the free hydroxyl ions and hence an increase in pH. The way the question is formulated only really makes sense if one assumes that the egg white acquired extra carbonic acid to reduce the alkalinity resulting from the calcium carbonate/hydroxide.