Analytic a posteriori claims are generally considered something of a paradox. First, let's recall that an analytic proposition's truth is entirely a function of its meaning -- "all widows were once married" is a simple example; certain claims about mathematical objects also fit here ("a pentagon has five sides.")
So, an analytic proposition is 'inherent' in a way that isn't the case for a synthetic proposition. Consider Kant's own example of a synthetic proposition: "all bodies are heavy." The reason this isn't analytic is that the predicate ("heavy") isn't 'contained' in the subject ("body"), as it would be for the claim, e.g., that a square is four-sided. There is an inherent 'ease' with analytic claims, since all one needs to do to know it is "extract" the predicate from the subject.
Now, the a priori/a posteriori distinction is about whether we know something from experience. This may seem similar but it is distinct from the analytic-synthetic question (which again is about whether the subject contains the predicate or not); note that many a priori claims are also synthetic. Perhaps the clearest examples of a priori claims are mathematical expressions (
Finally, let's consider the problematic hybrid you have asked after. A proposition that's analytic a posteriori would contain the predicate within the subject (as 'triangle' contains 'three sides') but would only be justifiable based on experience. Kant thought this category was paradoxical, as he thinks you never need to resort to experience to justify analytic claims.
However, some modern critics like Stephen Palmquist have argued that in fact philosophy requires these aposterior analytic claims to function in its characteristic 'hypothetical' mode:
To begin with, the impossibility of analytic a posteriori knowledge is generally considered to be 'quite evident' [P5:182-3]: indeed, it is a nonsensical contradiction in terms for those who equate 'analytic' and 'a priori' [see Ap. IV]. Even though Kant argues against those who identify analyticity and apriority [e.g., in Kt1:1-10], he joins them in dismissing this class of knowledge with only a brief explanation: 'it would be absurd to found an analytic judgment on experience. Since, in forming the judgment, I must not go outside my concept, there is no need to appeal to the testimony of experience in its support' [Kt1:11; cf. Kt2:268 and Kt4:12]. There are, however, a few theorists who do regard the analytic a posteriori as providing the best description of certain types of knowledge. Notwithstanding Kant's lack of concern for this class of knowledge, I shall argue in IV.3 that certain aspects of his philosophy can best be understood by reinterpreting them in terms of the analytic a posteriori. At this point, though, it will suffice to say that we should expect such knowledge, if it is possible, to have its validity grounded in some way in experience (a posteriori), and yet also to proceed by making inferences solely on the (analytic) basis of an application of the laws of logic to the concepts or propositions involved.
You can read Palmquist's whole book here. (This section appears in Chapter Four.)