The Case for AxI
The manga is much greater in scope than the anime; it spans over seven tankoubon and shows no sign of ending soon as of this writing, whereas the anime only covers material in the first two.
The manga version of events paralleled in the anime is notably lighter; there's more humor and less of the subtext-laden intensity that characterizes the dialog between these two. However, the manga goes on to become much more epic than the anime: globally, historically, and plot-wise its range is wider.
Especially in recent volumes, hints of AxI in the manga that are as direct as any in the anime have begun to surface. (This evidence is in chronological order, so the later exhibits may have spoilers depending on how much of the manga you've read. You have been warned.)
Remember to read right to left! (All scans and text are scanslations unless otherwise noted.)
Exhibit 0 (Other work by Hirano-san)
The infamous "grope drawing," the definitive piece of AxI-oriented official art, as pictured on a phone card.
Notice what Alucard's up to, and how Integra isn't mad about it. Nor is she looking submissive, passive, or in any way like her personality has been compromised. Rather, her icy look is turned on the viewer, calm and cool as ever regardless of the circumstances. She could very well be saying, "Yes, I'm shagging my vampire. You got a problem with that?"
This is, I think, the ultimate template for what AxI should be like.
Exhibit 1 (Volume 2, chapter 6)
This takes place in the National Gallery, under similar circumstances to the anime scene (the difference is that in the manga Enrico gives Integra information which spurs the plot to start unfolding; this is actually the point where manga and anime diverge). Again, Enrico calls Integra swine; again, Alucard enters and threatens Enrico's life because of it. Roughly the same rationale as that in Exhibit 4 of the anime evidence applies.
One difference, though: This time Walter's there. If Integra needs someone to smack Enrico around a bit, her Angel of Death is already on hand. So Alucard's appearance can't be rationalized as Integra needing to bring in somebody tough. He shows up because he wants to be there personally when her honor is insulted.
Exhibit 2 (Volume 3, Chapter 1)
This is the scene in which Alucard, Seras, and Pip are preparing to head off to Rio for the first major confrontation with one of the Millennium Nazis. In order to solve the problem of Seras' problem with crossing water (being a weak fledgeling vampire, she can't just do it), the poor girl has been packed in a coffin.
Integra, coming in to inspect the results, makes this comment about Alucard's outfit. She doesn't actually compliment it - that would be too open - but she definitely notices. And Integra doesn't seem the type to talk about clothing.
Exhibit 3 (Volume 3, Chapter 4)
Having slaughtered all members of the police unit sent after him in the hotel in Rio, Alucard gives Integra a call, just to let her know what's up. He then adds, "Now Integra, give me an order."
She'd already given him one order - "Search and destroy" - but now, having started to carry the "destroy" part of it out with respect to unwary humans, he tests her. Pushes her. Presses to see if she will have the strength to carry it through. "Without a moment's regret I can massacre them," he says at one point, "because I am a monster. What about you?"
Integra doesn't move for a moment, then directs Walter to give her a cigar. After a brief internal struggle (the only outward sign of her distress is the cigar trembling between her teeth), she bites down on it and gives the order:
Slamming her hand down on the table, Integra delivers a wonderfully dramatic example of her strength, force of character, and willingness to make tough decisions. (Walter approves, as you can see.)
It's not just Walter who likes this speech, however; Alucard responds with delighted laughter and a declaration of approval with Integra's response to his pushing.
The text of that approval in the Dark Horse translation is interesting. The lines are given here as "Very good, that's the last cover" and then "Magnificent! One who is hit will stand up straighter!" - showing how much he likes her strong responses to trouble, even trouble that he's caused. Dark Horse, however, writes it as "Yes, that is the last fig leaf. How splendid! It prompts a stirring in my loins." Not only does he like it . . . it turns him on. (Compare this with Exhibit 7 of the anime evidence.)
Exhibit 4 (Volume 4, Chapter 3)
Alucard's just finished the fight with Alhambra, and the Hellsing members are preparing to return home; Alucard gives Integra another call.
She directs him to return home and give a formal report on what he's learned from drinking Alhambra's blood, but she also tells him some things that aren't necessary for him to know. You wouldn't hold this kind of conversation with a simple servant.
And then, of course, Alucard asks this question - "Did [the battle] excite you?". It's sort of like a follow-up to the last call - "Your strength turns me on; does mine turn you on?" In response, Integra flips out and snaps at him. Obviously he's touched a nerve.
Exhibit 5 (Volume 4, Chapter 5)
Just a little observation . . . Alucard does have some strains of human emotion left, though he doesn't admit (or perhaps even realize) it. On the plane home from Brazil he wakes up from a dream about Abraham Van Helsing and is baffled to find that there are tears streaking his cheeks.
Exhibit 6 (Volume 4, Chapter 5)
Another brief observation: the two can pull off their seamless appearance of a conflict-free master/servant dynamic in front of others. This scene is the first time they've seen each other in person since those two phone calls, making the teamwork all the more impressive.
There are futher examples in volumes 6 and 7, but I can't track them down just yet. They'll show up here soon.
Viewers are encouraged to volunteer their own eyewitness testimonies; these can be found at the Witness Stand.
This is the summation of evidence I've gathered from the manga. Combined with the anime evidence, the obvious conclusion: AxI is a reasonable, realistic, and even very likely pairing. The defense rests.
The Case Against
The Witness Stand