Probably.
Probably not. How do you define real?
This is a question that I'm pretty sure that most people with hair out of the ordinary get this question at least once. From what I hear, people with natural hair different from the majority around them get this problem too. I get it quite a bit, partly because I do actually wear wigs quite a bit, and because I currently have emerald green hair.
I have worn my hair as my natural hair, un-dyed and only marginally styled. Sometimes I have dyed and heavily styled real hair. Sometimes I have hair-extensions, so some of my hair is my own hair and some of it is synthetic or someone else's. Sometimes - quite often, actually - I wear a wig. I can totally understand the curiosity behind this, because often my hair is rather elaborate, unnaturally or unusually coloured or styled, and tell-tale signs like the hair-line and crown and such can be disguised by whatever ornaments and head-dresses I might have in my hair.
I personally feel that "Is that your REAL hair" is usually an appropriate question to ask if you are approaching out of genuine curiosity, not a mentality of "let us examine the freak-show"; the latter is something I find to be hugely judgemental and very rude. I know some people find these sorts of questions a bit rude in that they feel that it is not any of the asker's business, especially (but not only) those who want their unnatural hair to pass as natural hair. Personally, I quite like displaying the artifice of my outfits and styling because I'm proud of the skill and creativity that I have put it into it.
If I am wearing a wig, and say so, please don't be offended if you ask to see my real hair and decline. I might be wearing a wig because I'm having a bad hair day, I might think my real hair does not match my outfit, or I might just not want to go through the fuss of putting my wig back on because taking it on and off properly means taking off my hair-ornaments, wig, wig-cap and and unpinning my real hair, and putting it back on means re-doing all of that and quite probably re-styling the wig and ornaments. While I understand the curiosity, I feel that it is perfectly within my rights to just not want to show it.
Please do not touch my hair, or anyone else's without asking - this so very rude! I cannot see why anyone would find it appropriate to touch a stranger without asking. Quite a few people will probably say that you can't touch their hair, but some of us are OK with it. I will let friends and acquaintances touch or poke my clothes, textured nails or hair, but even then I am not very keen on people that aren't Raven, my sister, or my hair-dresser/manicurist touching me. Some of the more extroverted people may well allow you to touch their hair.
Someone has actually cut off a thick lock of my silver wig while I was on a bus. I was very angry indeed about this, as they just damaged my property. First of all, good quality wigs with nice soft fibres - either human or quality synthetic fibres - are rather expensive. Secondly, it was a substantial enough amount to show the wig-cap underneath, and I have had to sew in more wefts of hair to try and patch the gap - which has cost my money for the replacement hair and time spent sewing. Most of all, though, SOMEONE HAD SCISSORS OR A SHARP IMPLEMENT NEAR MY HEAD ON MOVING PUBLIC TRANSPORT, but I didn't even know about it until I got home and took off my wig and saw the damage.
Questions such as "how do you get your hair like that!" or "what kind of dyes do you use" are perfectly acceptable questions, as they are for the most part genuine interest (and the person asking might be wanting to do their hair like that themselves) and as such, if I have the time, and am not hugely in a hurry, I will usually explain.
For the most part, while I do sometimes feel that I am not there to satisfy someone's idle curiosity, at least I am interesting enough to spark it, and while I do not revel in the attention that I get, and often do wish that I didn't garner quite so much of it, I accept that as someone out of the ordinary, people will notice and will ask questions. Human curiosity is a good thing, and answering questions (carefully) helps foster understanding - I would much rather field the questions of the idly curious than the insults of the idly judgemental.