are tekiyos and brachos m'akeiv each other

The gemara (R"H 34b) writes  תקיעות וברכות של ראש השנה ויום הכפורים מעכבות זו את זו  Tos (33b towards the end of the long Tos) quotes that Rashi explained that this means the two mitzvos of shofar and tefilah must go hand in hand -- you can't be yotzei tekiyos outside the context of tefilah, and can't be yotzei tefilah unless you blow shofar during the davening.  

Tos disagrees and says this can't be the pshat.  Proof #1 - the din is that if you have a choice between going to a place where there is shofar blowing or a place that has a minyan and sha"tz for davening, the gemara says to go hear shofar since it is d'oraysa.  If you can't be yotzei shofar without tefilah, what good is that?  

Proof #2 - the gemara says if someone davened without having a shofar and then later in the day managed to get a shofar, he blows.  According to Rashi, shouldn't he have to daven again, since davening without a shofar doesn't count?

Tos therefore learns that the gemara means that the 3 brachos of malchiyos, zichronos, and shofaros are m'akeiv each other, i.e. if you only know one but not the others, you can't be yotzei tefilah.  Similarly, the different kolos of shofar are m'akeiv each other, i.e. if you only know how to blow a tekiya, for example, you are not yotzei blowing that kol alone multiple times.  But the two mitzvos have nothing to do with each other.

It's very hard to understand this shitas Rashi.  What does he do with those gemaros?

In R' Wahrman's sefer Oros Ymei HaRachamim he quotes a very Brisker-ish sevara from R' Leizer Silver.  Rashi agrees that of course you can be yotzei the mitzvah of shofar even if you don't daven, and you can be yotzei davening without a shofar.  But Rashi understood that gemara on 34b to mean that in addition to the independent chiyuv of shofar and the independent chiyuv of tefilah, there is a new composite chiyuv of tekiyos al seder ha'brachos that requires both elements.  

The gemaras that Tos quotes are talking about the independent chiyuvim of shofar and tefilah, not the composite chiyuv.  It's that new composite chiyuv cannot be fulfilled unless you have both elements.

This, he suggests, is why many have the practice of blowing shofar even in the middle of the silent amidah.  The tefilah would be a tefilah even without shofar, but you would lack the added kiyum of the composite chiyuv that requires both.