Christ as the Rock of Exod 17 (1 Cor 10:1-4)?

In 1 Cor 10:1-4, Paul makes the shocking claim that the rock from which the Israelites drank in desert (Exod 17) was Christ:

For I do not want you to be unaware, brethren, that our fathers were all under the cloud and all passed through the sea; and all were baptized into Moses in the cloud and in the sea; and all ate the same spiritual food; and all drank the same spiritual drink, for they were drinking from a spiritual rock which followed them; and the rock was Christ.
(1 Cor 10:1-4; NAS1995)

On what basis does Paul make such an identification? Is this merely fanciful exegesis? Is there any theological basis for such an identification?