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I want to know if only travel agents can book connecting flights of different airlines on a single ticket or is it something anyone can do. If so how?

Say there are two flights A and B operated by airlines A1 and B1.

Can I simply call Airline A1 and book ticket and then call B1 and book another ticket and ask them to "append it to the same ticket" as that of A1?

edit. For example: A1 is China Airlines from TPE to BKK, B1 is Jet airways from BKK to BOM.

The way a single ticket with flights on two different airlines works is that the entire ticket is issued by one of the airlines -- say, A1. The airlines then have an interline agreement that allows A1 to sell legs on B1's flight at the fares and conditions that B1 publishes.

(This is different from a codeshare agreement, where A1 has the right to sell seats on some of B1's airplanes but where A1 sets the fares and tickets them mostly like seats on native A1 flights).

Interlining agreements are much more common than codeshares and generally exist between most pairs of full-service airlines whose route networks meet, irrespective of alliances.

What travel agents have is a reseller agreement with A1 that allows them to resell the combined A1+B1 tickets that A1 issues. The travel agent's computers plug into A1's sales system (usually through an intermediary called a Global Distribution System) at a lower level than the airline's own website, so they can book such a combined ticket through it.

Even though A1 can sell combined tickets, they will usually not have designed their website to make such sales beyond alliance boundaries. That's a marketing decision; they don't want to confuse online customers with the opportunity to book connections on their competitors' flights -- and customers who do need complex connections typically use travel agents anyway.

What this means for you: If you can find a physical ticket counter for either airline A1 or airline B1, they will generally be able to sell you a ticket for your entire combined itinerary at the same fares that you could get the combined itinerary for at a travel agent (which might not always be the cheapest fare either of the two airlines will sell their half for alone). Whether they're willing to do so is a different matter, but if the airline keeps a manned ticket counter open in the first place I would think the chance is pretty good.

I'm not sure why you would want it to do it that way, though. You're not going to get the ticket significantly cheaper directly from the airline (if you don't use their online sales channel that doesn't do interlining); they will just keep the commission they would otherwise pay the travel agent, and possibly even add an issuing fee of their own. In fact, some airline ticket counters are technically just travel agencies that happen to be owned by an airline and carry their branding.

If you are looking for a general way to "stitch" any two arbitrary flights onto a single ticket, then this is not at all possible (not even to travel agents), because the fare conditions for a given flight may prohibit it from being stitched with some others.

If you have a particular itinerary in mind, just do a search for this itinerary on a search engine, and it will give you all "reasonable" options, including itineraries which consist of flights of different airlines but can be booked on one ticket (because the fare conditions allow it).

Can I simply call Airline A1 and book ticket and then call B1 and book
another ticket and ask them to "append it to the same ticket" as that
of A1?

No; only the issuing airline can modify a ticket (modulo the occasional agreement between two airlines which have a very close partnership). You would need to call airline A1 and ask them to change your booking, if the fare conditions allow it. But you don't want to do that anyway; just book both flights at the same times to avoid having to rebook.

You seem to also be under the impression that a simple way to avoid booking fees is to "call the airline". This is usually not the case, and in fact many (most?) airlines will charge additional service fees for bookings made through their telephone reservation center (see British Airways).

A ticket from airline A can not be appended to a ticket by airline B, they will always be two separate tickets and two separate reservations.

Airline A maybe able to link the two reservations together, if airline B is in the same alliance, but it is still two tickets, so you would not get protected from late flights and such as a single ticket would. But it may let airline B know of your late arrival slightly earlier.

Airline A maybe able to book the flight on airline B for you, then issue a single ticket (or vice versa).

Travel agents and OTAs, like Expedia, do have the capability to stitch together bookings across multiple carriers, but some of the cheaper fares do not allow this type of booking.

I don't know of a publicly available website that lets you stitch fares and airlines together.

When you are searching for cheap fares, you sometimes have to give up something, like protection on tight connections, or build in longer layovers, eating up vacation time.

The answer is YES, but only for certain airline combinations.

What determines if this is even possible is the type of Interline Agreement the airlines have and if it allows them to book each other's inventory.

Most 'major' carriers have interlines, but not always, AA and DL for instance. Some airlines don't have any at all.

While a multi-carrier itinerary (which is different from a ticket) is linked (the PNR's, not the tickets) when booked, there is no way to link anything after the fact.