<p>We study a global theory of affine maximal surfaces with singularities, which are called affine maximal maps and are defined by Aledo–Martínez–Milán. In this paper, we define a special subclass of such surfaces other than improper affine fronts, called <i>affine maxfaces</i>, and investigate their global properties with respect to certain notions of completeness. In particular, by applying Euclidean minimal surface theory, we show that “complete” affine maxfaces satisfy an Osserman-type inequality. Moreover, one can also observe that affine maxfaces are in a class that does not contain non-trivial improper affine fronts. We also provide examples of such surfaces which are related to Euclidean minimal surfaces.</p>